Nonlinear resonances and self-excited oscillations of a rotor caused by radial clearance and collision

This paper investigates oscillations in a flexible rotor system with radial clearance between an outer ring of the bearing and a casing by experiments and numerical simulations. The mathematical model considers the collisions of the bearing with the casing. The following phenomena are found: (1) Nonlinear resonances of subharmonic, super-subharmonic and combination oscillation occur. (2) Self-excited oscillation of a forward whirling mode occurs in a wide range above the major critical speed. (3) Entrainment phenomena from self-excited oscillation to nonlinear forced oscillation occur at these nonlinear resonance ranges. Moreover, this study analyzes periodic solutions of the mathematical model by the Harmonic Balance Method (HBM). As the results, the nonlinear resonances of subharmonic oscillation and its entrainment phenomenon can be explained theoretically by investigating the stability of the periodic solutions. The influence of the static force and the bearing damping on these oscillation are also clarified.     

Dynamic Response and Stability of a Rotating Asymmetric Shaft Mounted on a Flexible Base

This paper presents the dynamic response and stability of an asymmetric rotating shaft supported by a flexible base near the major critical speed and the secondary critical speed. In this system, the base is movable only in a direction transversal to the shaft. In the theoretical analysis, taking into account the effects of damping, the unstable vibrations near the major critical speed are mainly considered, and also the behavior of the forced oscillations near the major and secondary critical speeds is investigated. From the theoretical analysis, the unstable region is found to be divided into at most six subregions which depend on the mass of the base, the stiffness of the base, and the asymmetry of the shaft. In addition, the resonance curves near unstable subregions are calculated. It is found that there exist two shapes of resonance curves. In experiments, five types of response curves, which contained n unstable subregion (n = 1, 2, ¨, 5) near the major critical speed, were obtained by changing the mass of the base. It was ascertained that the theoretical results for the behavior near the major critical speed agreed quantitatively with the experimental results.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

Two-layer hydraulic falls over an obstacle

Motions in a forced channel flow of two contiguous homogeneous fluids of different constant densities and different thicknesses are considered. The total depth is finite and the upper surface is constrained to be planar (rigid lid approximation). The forcing is due to a bottom obstruction. The existence of a critical thickness ratio, obtained when the square of the thickness ratio is equal to the density ratio, leads to major differences with the one-layer case. The present study concentrates on this critical case. Moreover it is restricted to hydraulic falls, which are steady flows over an obstacle providing a transition between a subcritical and a supercritical flow. A weakly nonlinear analysis is performed. At leading order the problem reduces to a forced modified Korteweg–de Vries equation which can be integrated exactly. The weakly nonlinear results are validated by comparison with a numerical integration of the full governing equations. The numerical method is based on boundary integral equation techniques. The differences with the one-layer case are the existence of a second family of subcritical hydraulic falls when the thickness ratio is below critical, and the existence of supercritical hydraulic falls described by four parameters instead of three for all thickness ratios.     

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.     

Resonance,Parameter Estimation,and Modal Interactions in a Strongly Nonlinear Benchtop Oscillator

We study the vibrations of a strongly nonlinear, electromechanically forced, benchtop experimental oscillator. We consciously avoid first-principles derivations of the governing equations, with an eye towards more complex practical applications where such derivations are difficult. Instead, we spend our effort in using simple insights from the subject of nonlinear oscillations to develop a quantitatively accurate model for the single-mode resonant behavior of our oscillator. In particular, we assume an SDOF model for the oscillator; and develop a structure for, and estimate the parameters of, this model. We validate the model thus obtained against experimental free and forced vibration data. We find that, although the qualitative dynamics is simple, some effort in the modeling is needed to quantitatively capture the dynamic response well. We also briefly study the higher dimensional dynamics of the oscillator, and present some experimental results showing modal interactions through a 0:1 internal resonance, which has been studied elsewhere. The novelty here lies in the strong nonlinearity of the slow mode.     

Nonlinear Dynamics of a Beam on Elastic Foundation

The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.     

异步进动转子与限位器碰摩试验

利用限位器来限制储能飞轮实验转子的大幅度低频异步进动,设计了转子与限位器碰摩试验装置,研究转子的碰摩振动。分析了转子内表面碰摩力对转子运动的影响。转子与内置式限位器发生稳定的局部碰摩时,转子低频进动幅值不再增加,转子自转速度保持不变。碰摩转子的强迫振动在时域及频域都表现出了复杂性,碰摩冲击作为宽频激励,能够激励出转子-支承系统的第二模态正向进动。     

质量任意分布下的柔性转子过临界点时的瞬态响应

本文首先将Riccati传递矩阵法和正、逆回旋运动分解理论应用于有阻尼的分布质量转子的复特征问题计算，求出系统各阶正、逆回旋临界速度（也称临界点）及相应振型然后作者对广义阻尼模态理论作了引伸和发展，结合Bogoliubov-Mitropolskii渐近法，建立起一阶微分方程组，计算不平衡柔性转子分别在正、逆回旋下通过各阶临界点的非线性、非定常瞬态响应，还深入分析了转子相继通过两十分邻近的临界点时发生的耦合现象。     

An investigation on primary resonance phenomena of elastic medium based single walled carbon nanotubes

In this paper, the geometrically nonlinear free and forced oscillations of simply supported single walled carbon nanotubes (SWCNTs) are analytically investigated on the basis of the Euler–Bernoulli beam theory. The nonlinear frequencies of SWCNTs with initial lateral displacement are discussed. Equations have been solved using an exact method for free vibration and multiple times scales (MTS) method for forced vibration and some analytical relations have been obtained for natural frequency of oscillations. The numerical results reveal that the nonlinear free and forced vibration of nanotubes is effected significantly by both surrounding elastic medium and CNT aspect ratio.     

高速涡轮增压器轴承-转子系统不平衡响应及稳定性分析

汽车涡轮增压器广泛采用浮环轴承支承的小型轻质转子系统,以实现100 000~300 000 r/min的工作转速,提高发动机功率和动力性能,并降低燃油消耗和排放. 在此超高速工况下,动压油膜的强非线性作用和转子固有的不平衡效应使该系统呈现出复杂的动力学现象,其中油膜涡动、振荡、跳跃、倍周期分岔和混沌等非线性动力学行为对增压器的健康运转意义重大,因而备受关注. 本文作者从摩擦学动力学耦合的角度出发,基于流体动压轴承润滑理论和有限差分法计算非稳态油膜压力,结合达朗贝尔原理和传递矩阵法建立了转子离散化动力学方程,提出了一种由双油膜浮环支承的涡轮增压器转子系统动力学模型,并从转子轨迹、轴承偏心率、频谱响应、庞加莱映射和分岔特性等方面比较分析,描述了该非线性轴承-转子系统的不平衡效应及油膜失稳特征. 结果表明:转子一般在相对低速下作稳定的单周期不平衡振动,在高转速下其被油膜失稳引起的次同步涡动所抑制,但不平衡量的增加可阻碍转子以拟周期运动通向混沌运动的路径;适当不平衡补偿下,由于内、外油膜间交互的非线性刚度和阻尼作用,在油膜失稳区间之间的中高速区会出现适合增压器健康运转的稳定区间.      

Performance enhancement of wing-based piezoaeroelastic energy harvesting through freeplay nonlinearity

We investigate experimentally how controlled freeplay nonlinearity affects harvesting energy from a wing-based piezoaeroelastic energy harvesting system. This system consisits of a rigid airfoil which is supported by a nonlinear torsional spring (freeplay) in the pitch degree of freedom and a linear flexural spring in the plunge degree of freedom. By attaching a piezoelectric material (PSI-5A4E) to the plunge degree of freedom, we can convert aeroelastic vibrations to electrical energy. The focus of this study is placed on the effects of the freeplay nonlinearity gap on the behavior of the harvester in terms of cut-in speed and level of harvested power. Although the freeplay nonlinearity may result in subcritical Hopf bifurcations (catastrophic for real aircrafts), harvesting energy at low wind speeds is beneficial for designing piezoaeroelastic systems. It is demonstrated that increasing the freeplay nonlinearity gap can decrease the cut-in speed through a subcritical instability and gives the possibility to harvest energy at low wind speeds. The results also demonstrate that an optimum value of the load resistance exists, at which the level of the harvested power is maximized.     

Forced Oscillations of a System,Containing a Snap-Through Truss,Close to Its Equilibrium Position

The nonlinear dynamics of a two-degree-of-freedom mechanical system is considered. This system consists of a linear oscillator under the action of a time-periodic force and a snap-through truss, which acts as an absorber of the forced oscillations of the linear main system. The forced oscillations of the snap-through truss close to its equilibrium position are analyzed by the multiple scales method.     

Nonlinear vibration analysis by an extended averaged equation approach

The paper presents an extended averaged equation approach to the investigation of nonlinear vibration problems. The proposed method is applied to some free and self-excited oscillators, the Duffing's forced oscillators including main resonance, subharmonic resonance and super harmonic resonance. The results in analyzing the vibration systems with arbitrary non-linearity show advantages of the method.     

Elastic Vibrations of a Single-Support Thin-Walled Rotor (Compound Shell) During Complex Rotation

Steady precession vibrations of a single-support thin-walled rotor whose rotation axis is forced to make an additional rotation are simulated numerically. It is established that compared with double-support rotors, single-support rotors are more dynamically compliant and undergo two precession resonances over the range of rotation speeds being considered. Mode shapes of precession vibrations are drawn     

Turbocharger rotor dynamics with foundation excitation

To investigate the effect of foundation excitation on the dynamical behavior of a turbocharger, a dynamic model of a turbocharger rotor-bearing system is established which includes the engine’s foundation excitation and nonlinear lubricant force. The rotor vibration response of eccentricity is simulated by numerical calculation. The bifurcation and chaos behaviors of nonlinear rotor dynamics with various rotational speeds are studied. The results obtained by numerical simulation show that the differences of dynamic behavior between the turbocharger rotor systems with/without foundation excitation are obviously. With the foundation excitation, the dynamic behavior of rotor becomes more complicated, and develops into chaos state at a very low rotational speed.     

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

FORCED VIBRATIONS WITH INTERNAL RESONANCE OF A PIPE CONVEYING FLUID UNDER EXTERNAL PERIODIC EXCITATION

Applying the multidimensional Lindstedt-Poincaré (MDLP) method, we study the forced vibrations with internal resonance of a clamped-clamped pipe conveying fluid under external periodic excitation. The frequency-amplitude response curves of the first-mode resonance with internal resonance are obtained and its characteristics are discussed; moreover, the motions of the first two modes are also analyzed in detail. The present results reveal rich and complex dynamic behaviors caused by internal resonance and that some of the internal resonances are decided by the excitation amplitude. The MDLP method is also proved to be a simple and efficient technique to deal with nonlinear dynamics.     

Forced nonlinear oscillations of semi-infinite cables and beams resting on a unilateral elastic substrate

In this work, we study the nonlinear oscillations of mechanical systems resting on a (unilateral) elastic substrate reacting in compression only. We consider both semi-infinite cables and semi-infinite beams, subject to a constant distributed load and to a harmonic displacement applied to the finite boundary. Due to the nonlinearity of the substrate, the problem falls in the realm of free-boundary problems, because the position of the points where the system detaches from the substrate, called Touch Down Points (TDP), is not known in advance. By an appropriate change of variables, the problem is transformed into a fixed-boundary problem, which is successively approached by a perturbative expansion method. In order to detect the main mechanical phenomenon, terms up to the second order have to be considered. Two different regimes have been identified in the behaviour of the system, one below (called subcritical) and one above (called supercritical) a certain critical excitation frequency. In the latter, energy is lost by radiation at infinity, while in the former this phenomenon does not occur and various resonances are observed instead; their number depends on the statical configuration around which the system performs nonlinear oscillations.     

Adaptive neural DSC for stochastic nonlinear constrained systems under arbitrary switchings

This paper focuses on the primary resonance analysis of a dual-rotor system having two rotor unbalance excitations of different rotating speeds and being connected by an inter-shaft ball bearing. Due to the complex excitation condition and the complicated nonlinear bearing forces of the inter-shaft bearing, the general analytical methods, e.g., the multiple scales method or the harmonic balance method, are failed to give the theoretical solutions. Thus, the harmonic balance–alternating frequency/time domain (HB–AFT) method is formulated to deal with this problem. The basic idea of the method is using the inverse discrete Fourier transform and the discrete Fourier transform, instead of the direct analytical relationship between the supposed solutions of the system and the nonlinear forces, to construct the harmonic expressions of the nonlinear forces, which is the so-called alternating frequency/time domain technique. By using the HB–AFT method, therefore, a Newton– Raphson iteration procedure can be performed to demonstrate the approximate solutions of the system. Accordingly, the frequency responses of the system affected by some critical parameters, such as rotating speed ratio, unbalances of both the inner and outer rotors, and clearance of the inter-shaft bearing, are analyzed, respectively. A variety of phenomena including double resonance peaks, biperiodic and quasi-periodic behaviors, and resonance hysteresis phenomenon are obtained, which are discussed in detail through diagrams for separated frequency responses with different frequency components and rotors’ orbits for different combinations of system parameters. Most prominently, for a relatively small unbalance of rotor as well as a relatively large clearance of the inter-shaft bearing, the resonance hysteresis phenomena are more obvious. The results obtained are also compared with the direct numerical simulation results, and the comparisons show good agreements. In addition, the methodology formulated in this paper is a general approach, which can be applied to other engineering systems with multi-frequency excitations.