Influence of viscous damping on friction induced travelling waves

We investigate the influence of viscous damping on the shape and stability of travelling waves induced by Coulomb friction between a rotating rigid shaft and an annular elastic cylinder. As expected, the damping causes the travelling wave solutions to become smoother and more stable. It also decreases the amplitude and range of separation solutions and may destroy the travelling waves by grazing bifurcations at large values of the damping parameters. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermoactive vibration control of shafts

Flexible long‐span rotating shafts exhibit flutter instability conditioned by internal friction in bending at high rotation speeds. Under usual working conditions a shaft may be additionally subjected to external excitations related to unbalance forcing or edge bearing movements. Flutter occurs at angular speeds exceeding the lowest natural frequency of the shaft as a nonrotating beam. Thus, externally excited resonances mostly appear in the subcritical speed zone although they can interact with flutter vibration as well. The present paper is concerned with resonant vibration control of shafts based on application of thermoactive SMA components in composite shaft structures, as conceptually shown in [1]. The well known unique properties of SMAs consisting in huge changes of the elastic modulus and its loss factor as results of a reversible martensitic phase transformation under slight temperature variations [2] promise to control shaft vibrations through temperature‐induced modal modifications of the structure. The main resonance of a simply supported rotating shaft is considered to be controlled by open‐loop SMA activation. Efficiency of thermoactive vibration control is analysed and a concept of an intelligent self‐controlled shaft structure is introduced. Geometric nonlinearity is assumed in modelling and computer simulations to show the thermoactive resonance suppression including the case when both externally excited and flutter vibrations interact.     

On Stability of a Thin‐Walled Shaft with Active Piezoelectric Fibers

The paper is concerned with the problem of active stabilisation of a rotating flexible shaft made of a composite material containing piezoelectric fibers being controllable by the applied electric field. Rotating shafts exhibit fluttertype instability while exceeding the critical angular velocity. The factor responsible for the loss of stability is internal friction present in the material of the shaft. In the case of a composite structure the internal friction is increased in comparison with steel shafts, and so is the susceptibility of the laminated shaft to self‐excitation. In the paper a method of stabilisation, i.e. shifting the critical threshold towards greater rotation speeds, possibly outside the operating range, is presented. The method is based on incorporation of piezoceramic fibers embedded into the host structure of the shaft. Such integral materials, reflecting the concept of a polymer matrix reinforced with active fibers, are known as Piezoelectric Fiber Composites (PFCs). The carried out examinations have proved that the method is efficient, however limited. It is shown that the critical rotation speed can be increased by several percents, but only within a certain range of structural parameters of the considered system.     

Hopf bifurcations of travelling waves in a brake-like system

We consider a system composed of an elastic tube, which is fixed at the outer boundary and in frictional contact with a rigid cylinder, rotating inside the tube about the common axis. Under the assumption of Coulomb's friction law at the contact surface between the two bodies several types of rotating slip-stick and also slip-stick-separation travelling waves with different wave numbers can be observed. For a wide range of parameters the linearized system has unstable complex eigenvalues, which cause high frequency oscillations and unpleasant squeal. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effect of dissipative and gyroscopic forces on the stability of a class of linear time-varying systems

It is shown that the effect of dissipative and gyroscopic forces on a certain class of potential linear time-varying system differs considerably from the effect of these forces on a time-invariant system. Examples are considered. In the first of these, the equations of motion of a disk, attached to a rotating weightless elastic shaft, are investigated, taking external friction into account. The results obtained are compared with the results obtained previously by others when considering this problem. In the second example, certain problems of the stability of rotation of a Lagrange top on a base subjected to vertical harmonic vibrations are investigated.     

Non-stationary analysis of a rotating shaft with geometrical nonlinearity during passage through critical speeds

In this paper, analysis of a rotating shaft with stretching nonlinearity during passage through critical speeds is investigated. In the model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. First, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with non-constant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the non-stationary vibration of the rotating shaft, the asymptotic method is applied to the equations expressed in normal coordinates. Two analytical expressions, as function of system parameters that describe the amplitude and phase of motion during passage through critical speeds are derived. The effects of angular acceleration, stretching nonlinearity, eccentricity and external damping on maximum amplitude of the shaft are investigated. It is shown that the nonlinearity has important effect on maximum amplitude when the rotating shaft passing through critical speeds, especially in low angular acceleration. To validate the results of the perturbation method, numerical simulation is applied.     

大位移Timoshenko转轴三维耦合动力学分析

对于高速柔性转轴,综合考虑滑移、弯曲、剪切变形、旋转惯性、陀螺效应和动不平衡等因素,运用Timoshenko旋转梁理论,给出弹性体空间运动的一般性描述,通过Hamilton原理建立弯曲-扭转-轴向三维耦合非线性动力学方程,应用参数摄动方法和假设振型方法进行化简,并用数值模拟分析了轴向刚性滑移、剪切变形、截面尺寸和转速等因素对转轴动力学响应的影响。     

Modelling of rotating shafts with flexible disks

This paper deals with the methodology of the modelling of rotating shafts with flexible disks. Rotating shafts are modelled as a one dimensional continuum on the basis of the Bernouli-Euler theory, which assumes that the shaft cross section remains a flat plane and is perpendicular to the centerline during vibration. Disks are modelled as a three dimensional continuum by means of the finite element method. The presented approach allows to introduce centrifugal and gyroscopic effects. The special coupling matrix is used for the connecting a rotating shaft and a mounted flexible disk. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mode Interaction at a Triple Zero Point of O(2) symmetric Nonlinear Systems with Two Parameters

§ 1.Introduction　Consider the nonlinear ODE with two parametersut+ g(u,λ,α) =0 (1 .1 )and its steady-state equationg(u,λ,α) =0 , (1 .2 )where g:U× R2 → U is a sufficiently smooth nonlinear mapping,and U a finitedimensional Hilbertspace.We remark thatitis easy to generalize ourresults to the casewhere g:U×R2→ V and U and V are infinite dimensional Hilbert spaces.So(1 .1 ) canbe e.g. a PDE. We assume that g is O(2 ) symmetric (equivalent,commutable) ,thatisγg(u,λ,α) =g(γu,…     

Effect of material in-homogeneity on electro-thermo-mechanical behaviors of functionally graded piezoelectric rotating shaft

In this article, a hollow circular shaft made from functionally graded piezoelectric material (FGPM) such as PZT_5 has been studied which is rotating about its axis at a constant angular velocity ω. This shaft subjected to internal and external pressure, a distributed temperature field due to steady state heat conduction with convective boundary condition, and a constant potential difference between its inner and outer surfaces or combination of these loadings. All mechanical, thermal and piezoelectric properties except for the Poisson’s ratio are assumed to be power functions of the radial position. The governing equation in polarized form is shown to reduce to a system of second-order ordinary differential equation for the radial displacement. Considering six different sets of boundary conditions, this differential equation is analytically solved. The electro-thermo-mechanical stress and the electric potential distributions in the FGPM hollow shaft are discussed in detail for the piezoceramic PZT_5. The presented results indicate that the material in-homogeneity has a significant influence on the electro-thermo-mechanical behaviors of the FGPM rotating shaft and should therefore be considered in its optimum design.     

Bifurcations of Stick-Slip-Separation Waves in a Brake-like System

We consider a system composed of an elastic tube in frictional contact with a rotating rigid cylinder. At the contact surface several types of rotating slip-stick and also slip-stick-separation waves with different wave numbers can be observed. In order to determine the stability of these travelling waves, we have to solve a non-local eigenvalue problem. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experiments on baroclinic instabilities

We study experiments on baroclinic instabilities in a different heated rotating annulus cooled from within. Characteristic time series of the flow velocity are measured with the Laser–Doppler–Velocimetry (LDV) technique. The surface flow is observed with a co–rotating camera. Steady waves as well as complex flow states are observed. Hysteresis is also found. The drift rate of stable baroclinic waves is determined. The methods of non–linear time series analysis are used to investigate the dynamics of baroclinic waves, particulary in transition zones between different flow regimes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodic Rotating Waves of a Geodesic Curvature Flow on the Sphere

Given a zone on the unit sphere S² with periodic undulating boundaries, we consider the motion of a curve in this zone which is driven by its geodesic curvature. First, we give a necessary and sufficient condition for the existence of periodic rotating waves. Then we study how the average rotating speed of the periodic rotating wave depends on the geometry of the boundaries. We find that when the period of the boundaries tends to 0, the homogenization limit of the rotating speed depends only on the maximum slope of the domain boundaries.     

迟滞型材料阻尼转轴的分岔

应用平均法研究迟滞型材料阻尼转轴的分岔.首先用Hamilton原理推导出复数形式的转轴运动微分方程,然后用平均法求出各阶模态主共振时的平均方程,并分析定常解的稳定性,最后用奇异性理论分析正常运动和失稳运动响应(异步涡动)的分岔.研究表明,一定参数条件下,转轴在通过各阶临界转速(主共振)时,可能会因受到冲击而失稳(Hopf分岔).正常运动响应在不平衡量较大时有滞后和跳跃现象,而失稳运动响应是一类余维数较高的非对称分岔.由于内阻尼的非线性,响应随转速增加时还可能产生二次Hopf分岔,对应原系统的双调幅运动.做好动平衡及提高外阻尼水平是避免这种大幅值自激振动的有效措施.     

On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model

This paper studies the wave propagation of two-dimensional functionally graded (2D-FG) porous rotating nano-beams for the first time. The rotating nano-beams are made of two different materials, and the material properties of the nano-beams alter both in the thickness and length directions. The general nonlocal theory (GNT) in conjunction with Reddy's beam model are employed to formulate the size-dependent model. The GNT efficiently models the dispersions of acoustic waves when two independent nonlocal fields are modelled for the longitudinal and transverse acoustic waves. The governing equations of motion for the 2D-FG porous rotating nano-beams are established using Hamilton's principle as a function of the axial force due to centrifugal stiffening and displacement. The analytic solution is applied to obtain the results and solve the governing equations. The effect of the features of different parameters such as functionally graded power indexes, porosity, angular velocity, and material variation on the wave propagation characteristics of the rotating nano-beams are discussed in detail.     

旋转功能梯度圆柱壳的固有频率计算

Love的一阶理论被用来计算旋转功能梯度圆柱壳的固有频率, 为了验证该方法的有效性,计算了简支不旋转各向同性和功能梯度圆柱壳的固有频率,计算结果与相关文献中的结果具有较好的一致性．通过几个数值算例,研究了幂指数、x和θ方向的波数、厚度-半径比对简支旋转功能梯度圆柱壳固有频率的影响．结果表明:后向波的固有频率随着转速的增加而增加,前向波的固有频率随着转速的增加而减小,前向波和后向波的固有频率随着厚度-半径比增加而增加．     

Finite element simulation for dynamic large elastoplastic deformation in metal powder forming

In this paper, a transient dynamic analysis of the powder compaction process is simulated by a large displacement finite element method based on a total and updated Lagrangian formulation. A combination of the Mohr–Coulomb and elliptical yield cap model, which reflects the stress state and degree of densification, is applied to describe the constitutive model of powder materials. A Coulomb friction law and a plasticity theory of friction in the context of an interface element formulation are employed in the constitutive modelling of the frictional behaviour between the die and powder. Finally, the powder behaviour during the compaction of a plain bush, a rotational flanged and a shaped tip component are analysed numerically. It is shown that the updated Lagrangian formulation, using a combination of the Mohr–Coulomb and elliptical cap model, can be effective in simulating metal powder compaction.     

Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

We investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction-advection-diffusion equations with periodic boundary conditions. Using zero number properties of the solutions and the phase shift equivariance of the equation, we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves.     

Effect of the rotation on a non-homogeneous infinite cylinder of orthotropic material

The present investigation is concerned with a study effect of non-homogeneous on the elastic stresses in rotating orthotropic infinite circular cylinder subjected to certain boundary conditions. Closed form stress solutions are obtained for rotating orthotropic cylinder with constant thickness for three cases: (1) a solid cylinder; (2) cylinder mounted on a circular rigid shaft; and (3) cylinder with a circular hole at the center. Analytical expressions for the components of the displacement and the stress in different cases are obtained. The effect of the rotation and non-homogeneity on the displacement and stress are studied. Numerical results are given and illustrated graphically for each case is considered. The effects rotating and non-homogeneity are discussed. Comparisons are made with the results predicted in the presence and absence of rotation.     

Transient heating of a rotating elastic-plastic solid shaft

Based on Tresca's yield criterion and the flow rule associated with it, the transient plasticization in a rotating elastic–plastic solid shaft with temperature dependent yield stress subject to a temperature cycle is studied. Special attention is paid to the residual stresses. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)