Nonlinear dynamic analysis of a rotor system with fixed-tilting-pad self-acting gas-lubricated bearings support

Based on the nonlinear theory, the unbalanced responses of a fixed-tilting-pad gas-lubricated journal bearing-rigid rotor system are investigated. A?time-dependent mathematical model is established to describe the pressure distribution of gas-lubricated journal bearing with nonlinearity. The rigid rotor supported by a fixed-tilting-pad self-acting gas-lubricated journal bearing is modeled. The differential transformation method has been employed to solve the time-dependent gas-lubricated Reynolds equation, and the dynamic motion equation has been solved by the direct integral method. The unbalanced responses of the rotor system supported by fixed-tilting-pad gas-lubricated journal bearings are analyzed by the orbit diagram, Poincaré map, time series, and spectrum diagram. The numerical results reveal periodic, period-3, and quasiperiodic motions of nonlinear behaviors of the system. Finally, the effects of pivot ratio and preload coefficient on the nonlinear dynamic characteristics of the fixed-tilting-pad self-acting gas-lubricated journal bearing-rotor system are investigated.     

Bifurcation and nonlinear dynamic analysis of a rigid rotor supported by two-lobe noncircular gas-lubricated journal bearing system

This paper presents the study of the dynamic analysis of a rigid rotor supported by a two-lobe non-circular gas-lubricated journal bearing. A finite element method has been employed to solve the Reynolds equation in static and dynamical states and the dynamical equations have been solved using Runge–Kutta method. To analyze the behavior of the rotor center in horizontal and vertical directions under the different operating conditions, the dynamic trajectory, the power spectra, the Poincare maps, and the bifurcation diagrams are used. Results of this study indicates that by considering bearing number and rotor mass as the parameters of the system, complex dynamic behavior comprising periodic, KT-periodic, and quasi-periodic responses of the rotor center has occurred.     

Continuation analysis of a nonlinear rotor system

Nonlinear Dynamics - Nonlinearities in rotating systems have been seen to cause a wide variety of rich phenomena; however, the understanding of these phenomena has been limited because numerical...     

Theoretical rotor dynamic analysis of two-pole induction motors regarding excitation due to static rotor eccentricity

The paper shows a theoretical rotor dynamic analysis of two-pole induction motors regarding excitation due to static rotor eccentricity, which causes an oscillating magnetic force in the air gap of the machine. The hereby caused rotor vibrations are mathematically described, showing the orbit movements of the rotor centre, the shaft journals and the bearing housings. The influence of the rotor speed as well as influence of the direction of the magnetic force on the vibration behavior is investigated, based on a numerical example. On the one hand the aim of the paper is to derive the mathematical coherences—based on a simplified rotor dynamic model—between rotor dynamics, electromagnetic and the oil film characteristics of the sleeve bearings for two-pole induction machines, considering electromagnetic excitation due to a static rotor eccentricity. On the other hand the aim is to derive a method for calculating the maximum height of the shaft vibrations—as a worst case—caused by a static rotor eccentricity. Therefore the paper shall prepare the basis for implementing this method into more detailed numerical programs, e.g., finite element programs.     

Consistent matrices in rotor dynamic

Summary The expressions for the consistent mass and gyroscopic matrices for a constant section shaft element are obtained taking into account both shear deformation and transversal inertia. The results are compared with closed form solutions, which are available in simple cases. The results obtained show that the study of the dynamic behaviour of the rotor with a model which includes rotational inertia but not shear deformation is, at least in the case examined, misleading. Formulae for matrix condensation and for taking into account the effects of axial load and of a linear unbalance distribution are given. Damped systems can be studied using the same model, provided that damping can be assumed to be of either viscous or hysteretic type. Some formulations found in the literature are however not considered correct. An application of consistent matrices to a model which includes damping and uses matrix condensation is shown.

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Sommario Nel presente lavoro vengono ricavate le espressioni della matrice congruente delle masse e giroscopica per un elemento di albero a sezione costante. Tali espressioni tengono conto sia della deformazione a taglio che dell'inerzia trasversale. I risultati, in termini di velocità critiche e di diagramma di Campbell, per un albero a sezione costante vengono confrontati con soluzioni in forma chiusa di varia complessità. Vengono discussi gli effetti della deformazione a taglio, dell'inerzia trasversale, dello smorzamento e dello sforzo assiale sul comportamento dinamico. Viene inoltre presa in considerazione una distribuzione lineare nota di squilibrio. Il modello cosi ottenuto può venire usato anche per lo studio di sistemi smorzati, purchè lo smorzamento possa essere considerato di tipo ?viscoso? o ?strutturale?. Viene infine mostrata un'applicazione ad un modello in cui si tiene conto anche dello smorzamento e che fa uso della condensazione delle matrici per ridurre l'ordine del problema dinamico.

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Nonlinear dynamic analysis of a rub-impact rotor supported by oil film bearings

A general model of a rub-impact rotor system is set up and supported by oil film journal bearings. The Jacobian matrix of the system response is used to calculate the Floquet multipliers, and the stability of periodic response is determined via the Floquet theory. The nonlinear dynamic characteristics of the system are investigated when the rotating speed and damping ratio is used as control parameter. The analysis methods are inclusive of bifurcation diagrams, Poincaré maps, phase plane portraits, power spectrums, and vibration responses of the rotor center and bearing center. The analysis reveals a complex dynamic behavior comprising periodic, multi-periodic, chaotic, and quasi-periodic response. The modeling results thus obtained by using the proposed method will contribute to understanding and controlling of the nonlinear dynamic behaviors of the rotor-bearing system.     

Nonlinear dynamic behaviors of a bolted joint rotor system supported by ball bearings

Archive of Applied Mechanics - A dynamic model of the rotor-bearing system with bolted joint structure is set up based on the Lagrange’s equations, which considers the bearing clearance, the...     

Rubbing analysis of a nonlinear rotor system with surface coatings

Taking into account surface coatings painted on disc and casing, a nonlinear rotor system is established for simulating unbalance-rubbing coupled fault. During the modeling process of shaft, the nonlinear geometric relation between displacement and strain is considered. According to the Hamilton's principle, the restoring force between disc and shaft is described by the equivalent spring and the equivalent damper, and then the equivalent dynamic model is further set up. For contact analysis, the novel contact force model is employed to describe the impact force between disc and casing. Meanwhile, the Coulomb model is used to simulate the frictional force. The motion equations of the equivalent dynamic model are solved numerically and the corresponding dynamic behaviors are analyzed by bifurcation diagram, whirl orbit and Poincaré map. The relationship between equivalent linear/nonlinear stiffness and structural parameters is analyzed in detail. Moreover, the effects of eccentricity of disc, surface coatings and radius of shaft on the dynamic characteristic of the equivalent model are discussed.     

Crack characteristic analysis of multi-fault rotor system based on whirl orbits

Nonlinear Dynamics - In this paper, the nonlinear crack characteristics of different multi-fault rotor system are analyzed for crack detection based on whirl orbits. The nonlinear crack response...     

Nonlinear behavior analysis of geared rotor bearing system featuring confluence transmission

In this paper, the nonlinear vibration characteristics of geared rotor bearing system and the interactions among gears, shafts, and plain journal bearings were studied. First, with the consideration of backlash, transmission error, time-varying mesh stiffness, and layout parameters, the dynamic model of geared rotor bearing system featuring confluence transmission was proposed. The nonlinear oil-film forces were computed with the Reynolds equation for finite-length journal bearings. Second, the responses of meshing vibration and bearing vibration were discussed. The numerical results revealed that the system exhibited a diverse range of periodic, sub-harmonic, and chaotic behaviors. Under different ranges of rolling frequency, the system got into chaos state through different roads. Moreover, in lower frequency, meshing vibration showed coexist of different periodic motions. Lastly, couplings of nonlinear oil-film force and nonlinear gear mesh force were discussed through a range of rolling frequencies. Gear-bearing dynamic interactions were demonstrated through the analysis of dynamic gear loads and dynamic bearing loads, and the coupling effect behaved different when rolling frequency changed.     

转子系统的平稳／非平稳随机地震响应分析

应用虚拟激励法结合精细时程积分计算了转子系统受平稳／非平稳随机地震激励的动力响应。采用虚拟激励分析将平稳随机激励转化为稳态简谐激励，将非平稳随机激励转化为瞬态确定性激励，即使对于非对称的油膜刚度阵和阻尼阵，算法仍然简单高效，并得到精确的结果。     

Rigid rotor dynamic stability using Floquet theory

The dynamic behaviour of a rigid rotor elastically connected by a constant speed joint is investigated. The effects of inertial inequalities and stiffness inequalities are evaluated and the stability characteristics are worked out. The Floquet theory is extensively applied to this system in order to obtain the stability limit curves. Comparisons with experimental results obtained by other authors, available in the literature, are reported in order to verify the procedure. Results concerning the unsymmetrical rotors are widely shown through parameters values restricted to realistic cases.     

任意高差的多档输电线动刚度理论建模

针对目前多档输电线动力特性研究中存在的不足,将任意一档导线视为子结构并基于现有的索振动理论获得了其动张力;利用相邻两档导线的动张力建立了悬垂绝缘子串绕其悬挂点摆动的动力学方程;综合每一档导线的振动方程以及悬垂绝缘子串的摆动方程建立了两档及三档输电线的动力学方程,求解得到由端部位移激励产生的动刚度;通过对动刚度进行分析得到了多档输电线的固有频率以及模态的理论表达形式;建立了小高差的两档及三档导线的有限元模型,理论分析结果与有限元结果一致,验证了理论公式的正确性。本文提出的动刚度理论对于研究多档输电线的动态响应具有重要的应用价值。     

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.     

Response and bifurcation analysis of a MDOF rotor system with a strong nonlinearity

A new HB (Harmonic Balance)/AFT (Alternating Frequency Time) method is further developed to obtain synchronous and subsynchronous whirling response of nonlinear MDOF rotor systems. Using the HBM, the nonlinear differential equations of a rotor system can be transformed to algebraic equations with unknown harmonic coefficients. A technique is applied to reduce the algebraic equations to only those of the nonlinear coordinates. Stability analysis of the periodic solutions is performed via perturbation of the solutions. To further reduce the computational time for the stability analysis, the reduced system parameters (mass, damping, and stiffness) are calculated in terms of the already known harmonic coefficients. For illustration, a simple MDOF rotor system with a piecewise-linear bearing clearance is used to demonstrate the accuracy of the calculated steady-state solutions and their bifurcation boundaries. Employing ideas from modern dynamics theory, the example MDOF nonlinear rotor system is shown to exhibit subsynchronous, quasi-periodic and chaotic whirling motions.     

Bifurcation analysis for nonlinear multi-degree-of-freedom rotor system with liquid-film lubricated bearings

The oil-film oscillation in a large rotating machinery is a complex high-dimensional nonlinear problem. In this paper, a high pressure rotor of an aero engine with a pair of liquid-film lubricated bearings is modeled as a twenty-two-degree-of-freedom nonlinear system by the Lagrange method. This high-dimensional nonlinear system can be reduced to a two-degree-of-freedom system preserving the oil-film oscillation property by introducing the modified proper orthogonal decomposition (POD) method. The efficiency of the method is shown by numerical simulations for both the original and reduced systems. The Chen-Longford (C-L) method is introduced to get the dynamical behaviors of the reduced system that reflect the natural property of the oil-film oscillation.     

Dynamic analysis of the variable stiffness support rotor system with elastic rings

Nonlinear Dynamics - Elastic rings are common rotor supporting structure, which have been widely used in aeroengine rotor support system. However, large inertia force and gyroscopic moment may...