Non-smooth stability analysis of the parametrically excited impact oscillator

The aim of this paper is to give a Lyapunov stability analysis of a parametrically excited impact oscillator, i.e. a vertically driven pendulum which can collide with a support. The impact oscillator with parametric excitation is described by Hill's equation with a unilateral constraint. The unilaterally constrained Hill's equation is an archetype of a parametrically excited non-smooth dynamical system with state jumps. The exact stability criteria of the unilaterally constrained Hill's equation are rigorously derived using Lyapunov techniques and are expressed in the properties of the fundamental solutions of the unconstrained Hill's equation. Furthermore, an asymptotic approximation method for the critical restitution coefficient is presented based on Hill's infinite determinant and this approximation can be made arbitrarily accurate. A comparison of numerical and theoretical results is presented for the unilaterally constrained Mathieu equation.     

An extensive FPGA-based realization study about the Izhikevich neurons and their bio-inspired applications

This study is aimed at analysing damping and gyroscopic effects on the stability of parametrically excited continuous rotor systems, taking into account both external (non-rotating) and internal (rotating) damping distributions. As case-study giving rise to a set of coupled differential Mathieu–Hill equations with both damping and gyroscopic terms, a balanced shaft is considered, modelled as a spinning Timoshenko beam loaded by oscillating axial end thrust and twisting moment, with the possibility of carrying additional inertial elements like discs or flywheels. After discretization of the equations of motion into a set of coupled ordinary differential Mathieu–Hill equations, stability is studied via eigenproblem formulation, obtained by applying the harmonic balance method. The occurrence of simple and combination parametric resonances is analysed introducing the notion of characteristic circle on the complex plane and deriving analytical expressions for critical solutions, including combination parametric resonances, valid for a large class of rotors. A numerical algorithm is then developed for computing global stability thresholds in the presence of both damping and gyroscopic terms, also valid when closed-form expressions of critical solutions do not exist. The influence on stability of damping distributions and gyroscopic actions is then analysed with respect to frequency and amplitude of the external loads on stability charts in the form of Ince–Strutt diagrams.

     

多自由度参激系统稳定性的数值分析

提出多自由度周期参激系统稳定性的数值直接法。通过将扰动方程表示成状态方程形式,再根据Flo-quet理论将扰动解表示成指数特征分量与周期分量之积,并将其周期分量与系统周期系数展成Fourier级数,导出一系列代数方程,建立矩阵特征值问题,从而由数值求解特征值可直接确定参激系统的稳定性。该方法可用于一般周期参激阻尼系统,特征值矩阵不含逆子阵。应用于斜拉索在支座周期运动激励下的参激振动不稳定性分析,数值结果表明该方法的有效性。     

Multi-frequency excitation of magnetorheological elastomer-based sandwich beam with conductive skins

The present work deals with the dynamic stability of a symmetric sandwich beam with magnetorheological elastomer (MRE) embedded viscoelastic core and conductive skins subjected to time varying axial force and magnetic field. The conductive skins induce magnetic loads and moments under the application of magnetic field during vibration. The MRE part works in shear mode and hence the dynamic properties of the sandwich beam can be controlled by magnetic fields due to the field dependent shear modulus of MRE material. Considering the core to be incompressible in transverse direction, classical sandwich beam theory has been used along with extended Hamilton's principle and Galarkin's method to derive the governing equation of motion. The resulting equation reduces to that of a multi-frequency parametrically excited system. Second order method of multiple scales has been used to study the stability of the system for simply supported and clamped free sandwich beams. Here the experimentally obtained properties of magnetorheological elastomers based on natural rubber have been considered in the numerical simulation. The results suggest that the stability of the MRE embedded sandwich beam can be improved by using magnetic field.     

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.     

陀螺效应对转子横向振动的影响分析

举例说明了在动力转子系统中陀螺效应对实际模型的影响。着重分析了转子陀螺效应对进动角速度、振型以及临界角速度的影响。并应用状态空间法求解陀螺系统的本征值问题。数值结果表明，在一些工程问题中，陀螺力对于转子系统振动特性的影响是不能忽略的。     

Stability of parametric vibrations of hybrid composite plates

In this paper, the dynamic stability of laminated hybrid composite plates subjected to periodic uniaxial stress and bending stress is studied. The governing equations of motion of Mathieu-type are established by using the Galerkin method with reduced eigenfunctions transforms. Based on Bolotin's method the regions of dynamic instability of laminated hybrid composite plates are determined by solving the eigenvalue problems. The effects of layer thickness ratio, layer number, core material and load parameter on the dynamic instability of laminated hybrid composite plates are investigated and discussed.     

Precessional vibrations and resonances of compound shells during complex rotation

A study is made of the problem of precessional vibrations excited in a thin-walled rotor with a branching meridional section by gyroscopic inertial forces during the complex rotation of the rotor. Methods of solving the problem are proposed. It is established that resonant vibrations may be excited while the rotor is undergoing complex rotation. Ukrainian Transportation University, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 74–81, June, 1999.     

Stability and torsional vibration analysis of a micro-shaft subjected to an electrostatic parametric excitation using variational iteration method

In this article stability and parametrically excited oscillations of a two stage micro-shaft located in a Newtonian fluid with arrayed electrostatic actuation system is investigated. The static stability of the system is studied and the fixed points of the micro-shaft are determined and the global stability of the fixed points is studied by plotting the micro-shaft phase diagrams for different initial conditions. Subsequently the governing equation of motion is linearized about static equilibrium situation using calculus of variation theory and discretized using the Galerkin’s method. Then the system is modeled as a single-degree-of-freedom model and a Mathieu type equation is obtained. The Variational Iteration Method (VIM) is used as an asymptotic analytical method to obtain approximate solutions for parametric equation and the stable and unstable regions are evaluated. The results show that using a parametric excitation with an appropriate frequency and amplitude the system can be stabilized in the vicinity of the pitch fork bifurcation point. The time history and phase diagrams of the system are plotted for certain values of initial conditions and parameter values. Asymptotic analytically obtained results are verified by using direct numerical integration method.     

Three dimensional analysis of the dynamic stability of piezoelectric circular cylindrical shells

Based on a three-dimensional theory, a set of Mathieu–Hill equations have been derived for the dynamic stability analysis of piezoelectric circular cylindrical shells subjected to combined periodic axial compression and radial electric field loading. Bolotin's method is employed to determine the dynamic instability regions. Obtained results show that both the piezoelectric effect and the electric field only have minor effect on the unstable region. On the contrary, the geometric parameters, the rigidity of constituent materials and the external loading play dominating role in the dynamic stability of piezoelectric shells.     

有约束阻尼的非保守力学系统的稳定性

利用Ｌｙａｐｕｎｏｖ直接法，研究了有势力、陀螺力、Ｒａｙｌｅｉｇｈ阻尼和约束阻尼同时作用的非线性非保守力学系统的稳定性．假设陀螺力依赖某参数ｈ，得到系统渐近稳定的两个定理．     

Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control

Non-linear feedback control provides an effective methodology for vibration mitigation in non-linear dynamic systems. However, within digital circuits, actuation mechanisms, filters, and controller processing time, intrinsic time-delays unavoidably bring an unacceptable and possibly detrimental delay period between the controller input and real-time system actuation. If not well-studied, these inherent and compounding delays may inadvertently channel energy into or out of a system at incorrect time intervals, producing instabilities and rendering controllers’ performance ineffective. In this work, we present a comprehensive investigation of the effect of time delays on the non-linear control of parametrically excited cantilever beams. More specifically, we examine three non-linear cubic delayed-feedback control methodologies: position, velocity, and acceleration delayed feedback. Utilizing the method of multiple scales, we derive the modulation equations that govern the non-linear dynamics of the beam. These equations are then utilized to investigate the effect of time delays on the stability, amplitude, and frequency–response behavior. We show that, when manifested in the feedback, even the minute amount of delays can completely alter the behavior and stability of the parametrically excited beam, leading to unexpected behavior and responses that could puzzle researchers if not well-understood and documented.     

Transient response of single-disk rotor on anisotropic supports passing through critical speed

This paper clarifies the nonlinear and nonstationary characteristics of transient vibration of a single-disk rotor system on anisotropic supports passing through critical speed under limited energy supply. By analytical dynamics and Bogoliubov-Mitropolskii asymptotic method, the governing equations of motion of the system are reduced to a set of first order differential equations capable of numerical integral solution. The influences of gyroscopic effect of the rotor, external and internal damping as well as the initial phase angle of static unbalance on transient response are discussed.     

Steady-state dynamics of a non-ideal rotor with internal damping and gyroscopic effects

A rotor driven by an ideal source, i.e., a source capable of delivering unlimited amount of power, becomes unstable beyond a certain threshold spin speed due to non-conservative circulatory forces. The circulatory forces considered in this paper arise out of rotating internal damping. If the drive is non-ideal then the rotor spin speed cannot exceed the stability threshold. This phenomenon is a type of the Sommerfeld effect. In this work, a DC motor driving four-degrees-of-freedom rotor with internal damping and gyroscopic effects is considered and the corresponding steady-state spin frequency and the whirl orbit amplitude are analytically derived as functions of the parameters of the drive and the rotor system.     

参数激励下非线性受控系统的动力学和稳定性

研究两自由度参数激励系统的非线性动力学与控制问题．利用Lagrange方程建立含反馈控制的参激捅及其驱动机构组成的系统动力学方程，以多尺度方法获得一阶近似控制方程．然后，对系统受一阶摸态参激主共振与一、二阶模态间3：1内共振联合作用下的幅额响应及其稳定性，以及反馈参数对系统稳态行为的影响作了详细分析．结果表明，响应的稳定域位置和大小取决于位移反馈，位移立方反馈改变了系统的非线性程度，速度反馈类似于阻尼，可使系统呈现自激振动特性．     

Fault diagnosis of a rotor bearing system using response surface method

Dynamic analysis of a high-speed rotor bearing systems is challenged by their highly nonlinear and complex properties. Hence, an approximate response surface method (RSM) is utilized to analyze the effects of design and operating parameters on the vibration signature of a rotor-bearing system. This paper focuses on accurate performance prediction, which is essential to the design of high performance rotor bearing system. It considers distributed defects such as internal radial clearance and surface waviness of the bearing components. In the mathematical formulation the contacts between the rolling elements and the races are considered as nonlinear springs, whose stiffnesses are obtained by using Hertzian elastic contact deformation theory. The governing differential equations of motion are obtained by using Lagrange's equations. In terms of the feature that the nonlinear bearing forces act on the system, a reduction method and corresponding integration technique is used to increase the numerical stability and decrease computer time for system analysis. Parameters effects are analyzed together and its influence considered with DOE and Surface Response Methodology are used to predict dynamic response of a rotor-bearing system.     

白噪声参激余维2分叉系统研究

为考查白噪声参激的具有非半简双零特征值的一类余维二分叉系统的样本稳定性并确定其首次分叉点的位置，本文使用Ｌ．Ａｒｎｏｌｄ的渐近分析方法研究了系统最大Ｌｙａｐｕｎｏｖ指数的渐近分析式．为进一步研究噪声对于此系统退化分叉的影响，本文将使用一维扩散过程的奇异边界理论，考查“隐藏在余维２分叉点之后”同宿分叉系统受参激白噪声影响的分叉行为．     

Effects of flexible support stiffness on the nonlinear dynamic characteristics and stability of a turbopump rotor system

Using a flexible support is an efficient approach to solving the subsynchronous problems in a turbopump. In this paper, nonlinear rotordynamic analysis of a liquid fuel turbopump with a flexible support is presented using a dynamic modeling including two key destabilizing factors, nonlinear hydrodynamic forces induced by seals and internal rotor damping. The methodology of the partitioned direct integration method (PDIM) is described for reducing the computational efforts efficiently. Combining the PDIM and the shooting method, a nonlinear stability analysis of the rotor system is performed effectively. The numerical results, which are in good agreement with test data, indicate that the effects of flexible support stiffness k on the dynamic characteristics and stability of the rotor system are significant. The first critical speed of the rotor system rises as a nonlinear function of k markedly. The second critical speed varies slightly and approximates a linear variation as k increases. The onset speed of instability of the rotor system rises initially and then reduces as k increases. The effect of seal nonlinearities at low k is contrary to that at high k and the effect of seal length on the system stability is more significant than that of seal radius. The results explain the nature of the subsynchronous motion of a turbopump rotor system with flexible support and can be used in the design and operation of a liquid fuel turbopump rotor system to eliminate its rotordynamic problem.     

控制力矩陀螺用主动电磁轴承数字控制系统设计与实现

讨论了以TMS320VC33为核心的数字控制系统硬件结构。针对磁悬浮控制力矩陀螺转子的大惯量、小跨距、扁平转子和高速旋转时的陀螺效应，采用积分分离和不完全微分PID加交叉反馈控制策略。实验证明，该系统能够有效地抑制高速转子的陀螺效应，达到控制力矩陀螺电磁轴承高速稳定运行的要求。     

Perturbation transfer matrix method for eigendata of one-dimensional structural system with parameter uncertainties

IntroductionThestructuralsystemswithparameteruncertaintiesareverycommoninengineering .AcomprehensivereviewonthesubjectofdynamicproblemsofthesesystemswaspresentedinRef.[1]byIbrahim .Theperturbationmatrixmethodplaysanimportantroleintheinvestigationofdynamic…