DYNAMIC ANALYSIS OF A SPATIAL COUPLED TIMOSHENKO ROTATING SHAFT WITH LARGE DISPLACEMENTS

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A simple spinning laminated composite shaft model

A simple spinning composite shaft model is presented in this paper. The composite shaft contains discrete isotropic rigid disks and is supported by bearings that are modeled as springs and viscous dampers. Based on a first-order shear deformable beam theory, the strain energy of the shaft are found by adopting the three-dimensional constitutive relations of material with the help of the coordinates transformation, while the kinetic energy of the shaft system is obtained via utilizing the moving rotating coordinate systems adhered to the cross-sections of shaft. The extended Hamilton’s principle is employed to derive the governing equations. In the model the transverse shear deformation, rotary inertia and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated. To verify the present model, the critical speeds of composite shaft systems are compared with those available in the literature. A numerical example is also given to illustrate the frequencies, mode shapes, and transient response of a particular composite shaft system.     

圆筒容器内起旋和降旋过程中分层液体混合现象

针对旋转圆筒容器内的两种分层流体,应用平面激光诱导荧光与高速摄影技术对互溶与不互溶液体进行了实验研究.结果表明,上下层液体密度梯度与黏度梯度方向是产生界面不稳定的关键因素.当二者的梯度方向相同时,起旋过程不会发生剧烈混合,降旋过程中轻流体会冲击重流体;当二者的梯度方向相反时,起旋过程中形成抽吸效应,其后期发生界面破碎,降旋过程中重流体冲击轻流体.在旋转的3个过程中,降旋过程对混合的作用最大,无论梯度方向是否相同,都会发生液体间的界面不稳定.     

Bending vibration analysis of rotating machineries using multibody dynamics technology — development of computer program RotB

This paper examines a computer program developed to analyze the vibration of rotating machineries based on theories of vibration and multibody dynamics (MBD). Bending vibration problems of rotating machineries have generally been categorized as either linear or nonlinear. Linear problems can be formulated by standard methods and nonlinear problems can be formulated by MBD methods. In our study, nonlinear problems are treated by the use of a general-purpose computer program, RecurDyn (RD). In the program we developed, rotor bending vibration analysis (RotB) structural properties such as shafts, rotating rotary disks, unbalanced masses and foundation structures are modeled as multibody elements. Also, nonlinearities such as contact, non-symmetrical shaft effects, bearing characteristics, nonlinear restoring and damping characteristics in the bearings are taken into account. The computational results demonstrate the validity of RotB.     

Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle

In the presented paper the equations of motion of a rotating composite Timoshenko beam are derived by utilising the Hamilton principle. The non-classical effects like material anisotropy, transverse shear and both primary and secondary cross-section warpings are taken into account in the analysis. As an extension of the other papers known to the authors a nonconstant rotating speed and an arbitrary beam’s preset (pitch) angle are considered. It is shown that the resulting general equations of motion are coupled together and form a nonlinear system of PDEs. Two cases of an open and closed box-beam cross-section made of symmetric laminate are analysed in details. It is shown that considering different pitch angles there is a strong effect in coupling of flapwise bending with chordwise bending motions due to a centrifugal force. Moreover, a consequence of terms related to nonconstant rotating speed is presented. Therefore it is shown that both the variable rotating speed and nonzero pitch angle have significant impact on systems dynamics and need to be considered in modelling of rotating beams.     

Contact analysis of a flexible bladed-rotor

This paper presents a model of fully flexible bladed rotor developed in the rotating frame. An energetic method is used to obtain the matrix equations of the dynamic behaviour of the system. The gyroscopic effects as well as the spin softening effects and the centrifugal stiffening effects, taken into account through a pre-stressed potential, are included in the model. In the rotating frame, the eigenvalues' imaginary parts of the latter matrix equation give the Campbell diagram of the system and its stability can be analysed through its associated eigenvalues' real parts. The turbo machine casing is also modelled by an elastic ring in the rotating frame through an energetic method. Thus, in some rotational speed ranges the contact problem between the rotor and the stator can be treated as a static problem since both structures are stationary to each other. Prior to the study of the complete problem of contact between the flexible blades of the rotor and the flexible casing, a simple model of an elastic ring having only one mode shape, excited by rotating loads is developed in the rotating frame too, in order to underline divergence instabilities and mode couplings. Then, the complete problem of frictionless sliding contact between the blades and the casing, without rubbing, is studied. The stable balanced static contact configurations of the structure are found as function of the rotational speed of the rotor. Finally, the results are compared to these of the simple model of rotating spring-masses on an elastic ring, showing good adequacy. The present model of rotor appears thus particularly adapted to the study of blades-casing contacts and highlighted an unstable phenomenon near the stator critical speed even in case of frictionless sliding.     

Transverse vibrations of rotating shafts: Probability density and first-passage time of whirl radius

Transverse vibrations are considered for a single mass/two-degrees-of-freedom rotating shaft with linear internal or “rotating” damping and nonlinear external damping. The shaft is excited by external random forces. Analysis of resulting random vibrations is based on stochastic averaging method which yields separated (in the linear approximation) equations for complex amplitudes of forward and backward whirling motions. The former of these motions is shown to be dominant at rotation speeds in the vicinity of the instability threshold. Using this approximation an analytical solution is obtained for probability density of squared radius of the shaft's whirl. This solution can be used to detect on-line shaft's instability from its observed response. Solution is also obtained for expected time for reaching given level by the squared whirl radius of the shaft.     

Hopf bifurcation analysis of asymmetrical rotating shafts

The Hopf and double Hopf bifurcations analysis of asymmetrical rotating shafts with stretching nonlinearity are investigated. The shaft is simply supported and is composed of viscoelastic material. The rotary inertia and gyroscopic effect are considered, but, shear deformation is neglected. To consider the viscoelastic behavior of the shaft, the Kelvin–Voigt model is used. Hopf bifurcations occur due to instability caused by internal damping. To analyze the dynamics of the system in the vicinity of Hopf bifurcations, the center manifold theory is utilized. The standard normal forms of Hopf bifurcations for symmetrical and asymmetrical shafts are obtained. It is shown that the symmetrical shafts have double zero eigenvalues in the absence of external damping, but asymmetrical shafts do not have. The asymmetrical shaft in the absence of external damping has a saddle point, therefore the system is unstable. Also, for symmetrical and asymmetrical shafts, in the presence of external damping at the critical speeds, supercritical Hopf bifurcations occur. The amplitude of periodic solution due to supercritical Hopf bifurcations for symmetrical and asymmetrical shafts for the higher modes would be different, due to shaft asymmetry. Consequently, the effect of shaft asymmetry in the higher modes is considerable. Also, the amplitude of periodic solutions for symmetrical shafts with rotary inertia effect is higher than those of without one. In addition, the dynamic behavior of the system in the vicinity of double Hopf bifurcation is investigated. It is seen that in this case depending on the damping and rotational speed, the sink, source, or saddle equilibrium points occur in the system.     

Finite-Element Dynamic Analysis of a Rotating Shaft with or without Nonlinear Boundary Conditions Subject to a Moving Load

A C ⁰ continuity isoparametricfinite-element formulation is presented for the dynamic analysis of arotating or nonrotating beam with or without nonlinear boundaryconditions subject to a moving load. The nonlinear end conditions arisefrom nonlinear rolling bearings (both the nonlinear stiffness andclearance(s) are accounted for) supporting a rotating shaft. The shaftfinite-element model includes shear deformation, rotary inertia, elasticbending, and gyroscopic effect. Lagrange's equations are employed toderive system equations of motion which, in turn, are decoupled usingmodal analysis expressed in the normal coordinate representation. Theanalyses are implemented in the finite-element program DAMRO 1.Dynamic deflections under the moving load of rotating and nonrotatingsimply supported shafts are compared with those obtained using exactsolutions and other published methods and a typical coincidence isobtained. Samples of the results, in both the time and frequencydomains, of a rotating shaft incorporating ball bearings are presentedfor different values of the bearing clearance. And the results show thatsystems incorporating ball bearings with tight (zero) clearance have thesmallest amplitude-smoothest profile dynamic deflections. Moreover, fora system with bearing clearance, the vibration spectra of the shaftresponse under a moving load show modulation of the system naturalfrequencies by a combination of shaft rotational and bearing cagefrequencies. However, for a simply supported rotating shaft, the firstnatural frequency in bending dominates the response spectrum. The paperpresents the first finite-element formulation for the dynamic analysisof a rotating shaft with or without nonlinear boundary conditions underthe action of a moving load.     

Nonsynchronous vibrations of rotor system induced by oil-block inside the rotating drum

Lubrication oil in a rotor system guarantees the rotating components working smoothly and protects the system from being damaged due to friction. A volume of lubrication oil, however, sometimes leaks into the inner cavity of shaft and drums of rotor system and forms an oil-block during rotating operation. The oil-block usually induces abnormal vibration of the rotating machine, which is often observed in practical cases, such as in aero-engine. The work in this paper studies the nonsynchronous vibration (NSV) induced by an oil-block in a rotating drum of a Jeffcott rotor system, which consists of a shaft, a drum and two supporting isotropic bearings. The additional effect due to an oil-block rotating on the inner wall of the drum is included into rotor system differential equations considering the Coriolis acceleration and friction interaction between the oil-block and the drum. Numerical simulations are carried out under two rotating speeds conditions: a lower one and a higher one than the first critical rotor speed, which are defined as rigid rotor case and flexible rotor case. Numerical results states the transverse vibrations by bifurcation diagrams, shaft center trajectories, frequency spectra and Poincare diagrams, which reveal multi-periodic, quasi-periodic and other complex motions due to the existing of oil-block. The internal friction coefficient and mass of the oil-block are found to have a significant effect on the generation and development of NSV. As the oil-block case is very common in practice, the investigation of NSV caused by oil-block in rotor system would benefit the understanding of complex phenomena and contribute to fault detection and diagnosis of rotating machine.     

Blowup of solutions for the planar motions of rotating nonlinearly elastic rods

This paper treats the motion of flexible, extensible, shearable nonlinearly elastic rods, described by a geometrically exact theory, when they are confined to a plane rotating about a fixed axis at constant angular speed and when they are confined to a fixed plane with one end rotating at a constant angular speed about an axis perpendicular to the fixed plane. The paper gives restrictions on the constitutive equations and initial conditions that ensure that motions become unbounded at rapid rates as time becomes infinite. The analysis of these constitutive restrictions employs the theory of characteristics for single first-order semilinear partial differential equations.     

On the dynamics of rings rotating with variable spin speed

This work investigates nonlinear dynamic response of circular rings rotating with spin speed which involves small fluctuations from a constant average value. First, Hamilton's principle is applied and the equations of motion are expressed in terms of a single time coordinate, representing the amplitude of an in-plane bending mode. For nonresonant excitation or for slowly rotating rings, a complete analysis is presented by employing phase plane methodologies. For rapidly rotating rings, periodic spin speed variations give rise to terms leading to parametric excitation. In this case, the vibrations that occur under principal parametric resonance are analyzed by applying the method of multiple scales. The resulting modulation equations possess combinations of trivial and nontrivial constant steady state solutions. The existence and stability properties of these motions are first analyzed in detail. Also, analysis of the undamped slow-flow equations provides a global picture for the possible motions of the ring. In all cases, the analytical predictions are verified and complemented by numerical results. In addition to periodic response, these results reveal the existence of unbounded as well as transient chaotic response of the rotating ring.     

Non-linear vibration of variable speed rotating viscoelastic beams

Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton’s principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.     

Grazing Bifurcation in the Response of Cracked Jeffcott Rotor

A cracked rotor is modeled by a piecewise linear system due to thebreath of crack in a rotating shaft. The differential equations ofmotion for the nonsmooth system are derived and solved with thenumerical integration method. From the simulation results, it isobserved that a grazing bifurcation exists in the response. Thegrazing bifurcation can give rise to jumps between periodic motions,quasi-periodic motions from the periodic ones, chaos, and intermittentchaos.     

Nonlinear vibrations of blade with varying rotating speed

This paper investigates the nonlinear dynamic responses of the rotating blade with varying rotating speed under high-temperature supersonic gas flow. The varying rotating speed and centrifugal force are considered during the establishment of the analytical model of the rotating blade. The aerodynamic load is determined using first-order piston theory. The rotating blade is treated as a pretwist, presetting, thin-walled rotating cantilever beam. Using the isotropic constitutive law and Hamilton??s principle, the nonlinear partial differential governing equation of motion is derived for the pretwist, presetting, thin-walled rotating beam. Based on the obtained governing equation of motion, Galerkin??s approach is applied to obtain a two-degree-of-freedom nonlinear system. From the resulting ordinary equation, the method of multiple scales is exploited to derive the four-dimensional averaged equation for the case of 1:1 internal resonance and primary resonance. Numerical simulations are performed to study the nonlinear dynamic response of the rotating blade. In summary, numerical studies suggest that periodic motions and chaotic motions exist in the nonlinear vibrations of the rotating blade with varying speed.     

具有非轴对称刚度转轴的分岔

研究具有非轴对称刚度转轴的１／２亚谐共振和分岔,首先用Ｈａｍｉｌｔｏｎ原理导出运动微分方程,这是刚度系数周期性变化的参数激励方程,然后用多尺度法求得平均方程分岔响应方程和定常解,最后用奇异性理论分析分岔响应方程和定常解的稳定性,得到了局部分岔集和不同区域的不同分岔响应曲线。     

非惯性系下柔性悬臂梁的振动主动控制

采用变结构控制方法对非惯性系下柔性悬臂梁的振动主动控制进行研究．重点通过算例揭示一次近似模型与传统的零次近似模型的巨大差异，以及变结构方法在控制非惯性系下柔性悬臂梁的稳态振动的有效性．结果表明，当大范围旋转运动角速度较大时，传统零次近似模型不能对动力系统进行正确的数学描述；变结构控制方法能够使得非惯性系下梁的稳态振动得到完全镇定，且该方法对转动角速度变化具有较好的鲁棒性；采用零次近似模型进行控制设计的控制效果将在某一临界角速度条件下出现失效，该临界角速度值大于静止悬臂梁的基频．     

Rotating sandwich cylindrical shells with an FGM core and two FGPM layers: free vibration analysis

The free vibration analysis of a rotating cylindrical shell with an analytical method is investigated. The shell is considered as a sandwich structure, where the middle layer is a functionally graded material(FGM) shell, and it is surrounded by two piezoelectric layers. Considering piezoelectric materials to be functionally graded(FG),the material properties vary along the thickness direction as one innovation of this study.Applying the first-order shear deformation theory(FSDT), the equations of motion of this electromechanical system are derived as the partial differential equations(PDEs) using Hamilton's principle. Then, the Galerkin procedure is used to discretize the governing equations, and the present results are compared with the previously published results for both isotropic and FGM shells to verify the analytical method. Finally, the effects of FGM and functionally graded piezoelectric material(FGPM) properties as well as the thickness ratio and the axial and circumferential wave numbers on the natural frequencies are studied. Moreover, the Campbell diagram is plotted and discussed through the governing equations. The present results show that increasing the non-homogeneous index of the FGM decreases the natural frequencies on the contrary of the effect of non-homogeneous index of the FGPM.     

Design and Simulation of a Novel Multi-Axial Fatigue Test Rig

This research covers the design and simulation of a novel experimental concept for multi-axial fatigue analysis of cylindrical specimens. The resulting design allows a combination of bending and torsional stress to test specimens with critical diameters ranging from 5 to 15 mm at test frequencies up to 50 Hz. Furthermore, the amplitude and frequency of both loadcases can be controlled independently. The test rig will be used to study and validate fatigue criteria for synchronous and asynchronous loading conditions and to analyze the effect of size on the fatigue life of metal and plastic components. The test setup consists of a closed mechanical loop. The primary shaft contains the cylindrical test specimen and is modified to impose rotating bending loads. The secondary shaft is adjusted to introduce fluctuating torque in the transmission loop. Both shafts are connected by means of two double link mechanisms to minimize the clearance and the inertia of the system. The time-varying multi-axial stress state in the cylindrical specimen is analyzed as a function of the amplitudes and frequencies of both bending and torsional loadcases. This is verified by a numerical fatigue analysis in MSC-Patran and MSC-Fatigue. Finally, the dynamical behavior of the test system is studied using a 5 DOF torsional mass-spring representation and the Lagrangian method. A more complex model with 20 DOF is implemented in SimDriveLine and solved via Matlab to analyze the kinematical and dynamical properties more accurately. Both studies take the mechanical properties of steel and plastic test specimens of different sizes into account.     

基于径向基点插值法的旋转Mindlin板高次刚柔耦合动力学模型

将无网格径向基点插值法(radial point interpolation method,RPIM)用于中心刚体?旋转柔性板的动力学分析.基于浮动坐标系方法和一阶剪切变形理论即Mindlin板理论,考虑剪切变形的影响,并计入板面内变形的非线性耦合变形项,采用径向基点插值法描述板的变形场,保留动能中有关非线性耦合变形项...