Stability analysis of an open cracked rotor with the anisotropic rotational damping in rotating operation

The damping effects with the distinction of stationary damping and the anisotropic rotating damping on the dynamic stability of the rotating rotor with an open crack on the surface of the shaft is studied. The motion equations of the cracked rotor system are formed by Lagranges principal. Different from previous studies, the anisotropic system with the multi periodical varied coefficients is simplified by the moving frame method such that the stability analysis based on the root locus method can be applied. The corresponding Campbell diagram, decay rate plot and roots locus plot are derived to prove the destabilizing influence of both the rotational damping and the varied anisotropy ratio of the rotating damping. The effects of anisotropy of stiffness on the decisions of the critical range are also presented. The result with theoretical precision would not only generally provide practical applicability to crack detection and instability control of the heavy loading turbo-machinery system, but also give the suggestion that, the increased proportion and the aggravated anisotropy of the rotational damping due to the crack of the fatigue rotor should been taken into consideration on the modeling of cracked rotor system.     

Steady-state response of a geared rotor system with slant cracked shaft and time-varying mesh stiffness

The dynamic behavior of geared rotor system with defects is helpful for the failure diagnosis and state detecting of the system. Extensive efforts have been devoted to study the dynamic behaviors of geared systems with tooth root cracks. When surface cracks (especially for slant cracks) appear on the transmission shaft, the dynamic characteristics of the system have not gained sufficient attentions. Due to the parametric excitations induced by slant crack breathing and time-varying mesh stiffness, the steady-state response of the cracked geared rotor system differs distinctly from that of the uncracked system. Thus, utilizing the direct spectral method (DSM), the forced response spectra of a geared rotor system with slant cracked shaft and time-varying mesh stiffness under transmission error, unbalance force and torsional excitations are, respectively, obtained and discussed in detail. The effects of crack types (straight or slant crack) and crack depth on the forced response spectra of the system without and with torsional excitation are considered in the analysis. In addition, how the frequency response characteristics change after considering the crack is also investigated. It is shown that the torsional excitations have significant influence on the forced response spectra of slant cracked system. Sub-critical resonances are also found in the frequency response curves. The results could be used for shaft crack detection in geared rotor system.     

飞行器内裂纹转子系统的非线性动力学特性研究

在Jeffcott转子的开闭裂纹及方波模型基础上,建立了飞行器内裂纹转子系统的运动模型．数值研究表明:当飞行器以不同的等速度飞行时,转子轴与水平面之间夹角的变化将造成重力分量的变化,从而使转子运动在周期解、拟周期或浑沌状态之间变化,而且出现非线性现象的转速比、刚度变化比等参数的范围、进入和退出浑沌的路径、响应中的频率成份也会发生变化．飞行器的飞行速度变化还会改变裂纹转子响应的稳定性．飞行器等速飞行后的加速过程将引起转子振幅的突升及其后的下降,而且会使裂纹转子系统响应可能在不同的非线性状态下交替改变．     

Implementation of a cohesive zone model for the investigation of the dynamic behavior of a rotating shaft with a transverse crack

The influence of a transverse crack on the vibration of a rotating shaft has been at the focus of attention of many researchers. The knowledge of the dynamic behavior of cracked shaft has helped in predicting the presence of a crack in a rotor. Here, the changing stiffness of the cracked shaft is investigated based on a cohesive zone model. This model is developed for mode-I plane strain and accounts for triaxiality of the stress state explicitly by using basic elastic-plastic constitutive relations. Then, the proposed numerical solution is compared to the switching crack model, which is based on linear elastic fracture mechanics. The cohesive zone model is implemented in finite element techniques to predict and to analyse the dynamic behavior of cracked rotor system. Timoshenko beam theory is used to model the discrete shaft under the effect of gravity, unbalance force and gyroscopic effect. The analysis includes the cohesive function for describing the breathing crack and the reduction of the second moment of area of the element at the location of the crack. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of crack in a rotor system based on wavelet finite element method

The dynamics and diagnosis of cracked rotor have been gaining importance in recent years. In the present study a model-based crack identification method is proposed for estimating crack location and size in shafts. The rotor system has been modeled using finite element method of B-spline wavelet on the interval (FEM BSWI), while the crack is considered through local stiffness change. Based on Rayleigh beam theory, the influences of rotatory inertia on the flexural vibrations of the rotor system are examined to construct BSWI Rayleigh beam element. The slender shaft and stiffness disc are modeled by BSWI Rayleigh–Euler beam element and BSWI Rayleigh–Timoshenko beam element, respectively. Then the crack identification forward and inverse problems are solved by using surface-fitting technique and contour-plotting method. The experimental examples are given to verify the validity of the BSWI beam element for crack identification in a rotor system. From experimental results, the new method can be applied to prognosis and quantitative diagnosis of crack in a rotor system.     

Free vibration analysis of cracked postbuckled plate

In this paper, a methodology is introduced to address the free vibration analysis of cracked plate subjected to a uniaxial inplane compressive load for the first time. The crack, assumed to be open and at the edge is modeled by a massless linear rotational spring. The governing differential equations are derived using the Mindlin theory, taking into account the effect of initial imperfection. The response is assumed to be consisting of static and dynamic parts. For the static part, differential equations are discretized using the differential quadrature element method and resulting nonlinear algebraic equations are solved by an arc-length strategy. Assuming small amplitude vibrations of the plate about its buckled state and exploiting the static solution in the linearized vibration equations, the dynamic equations are converted into a non-standard eigenvalue problem. Finally, natural frequencies and modal shapes of the cracked buckled plate are obtained by solving this eigenvalue problem. To ensure the validity of the suggested approach an experimental setup and a numerical finite element model have been made to analyze the vibration of a cracked square plate with simply supported boundary conditions. Also, several case-studies of cracked buckled plate problem have been solved utilizing the proposed method, and effects of selected parameters have been studied. The results show that the applied load and geometric imperfection as well as the position, size and depth of the crack have different impact on natural frequencies of the plate.     

Vibration analysis of cracked post-buckled beams

Vibration analysis of cracked post-buckled beam is investigated in this study. Crack, assumed to be open, is modeled by a massless rotational spring. The beam is divided into two segments and the governing nonlinear equations of motion for the post-buckled state are derived. The solution consists of static and dynamic parts, both leading to nonlinear differential equations. The differential quadrature has been used to solve the problem. First, it is applied to the equilibrium equations, leading to a nonlinear algebraic system of equations that will be solved utilizing an arc length strategy. Next, the differential quadrature is applied to the linearized dynamic differential equations of motion and their corresponding boundary and continuity conditions. Upon solution of the resulting eigenvalue problem, the natural frequencies and mode shapes of the cracked beam are extracted. Several experimental as well as numerical case studies are performed to demonstrate the effectiveness of the proposed method. The investigation also includes an examination of several parameters influencing the dynamic behavior of the problem. The results show that the position and size of the crack as well as the geometric imperfection and applied load largely affect the modal shapes and natural frequencies of the beam.     

Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending

A mathematical model incorporating the higher order deformations in bending is developed and analyzed to investigate the nonlinear dynamics of rotors. The rotor system considered for the present work consists of a flexible shaft and a rigid disk. The shaft is modeled as a beam with a circular cross section and the Euler Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. The kinetic and strain (deformation) energies of the rotor system are derived and the Rayleigh–Ritz method is used to discretize these energy expressions. Hamilton’s principle is then applied to obtain the mathematical model consisting of second order coupled nonlinear differential equations of motion. In order to solve these equations and hence obtain the nonlinear dynamic response of the rotor system, the method of multiple scales is applied. Furthermore, this response is examined for different possible resonant conditions and resonant curves are plotted and discussed. It is concluded that nonlinearity due to higher order deformations significantly affects the dynamic behavior of the rotor system leading to resonant hard spring type curves. It is also observed that variations in the values of different parameters like mass unbalance and shaft diameter greatly influence dynamic response. These influences are also presented graphically and discussed.     

Cohesive zone model for a transverse breathing crack in a rotor

The breathing mechanism of a transversely cracked rotor and its influence on a rotor system that appears due to the shaft weight is studied. This breathing mechanism is based on experimental and simulation result for the crack shape reported in the literature. If the crack depth is small, the crack closure line is a straight line while for larger crack depths the crack closure becomes more curved. For both cases, a method is proposed for the evaluation of the stiffness losses in the cross section that contains the crack. This method is based on a cohesive zone model (CZM) instead of linear elastic fracture mechanics (LEFM) approach, because LEFM is valid only for the fully open crack and cannot be extended to other intermediate situations. As the crack is closed, the stress intensity factor (SIF) will not appear at the boundary between the closed cracked areas and the open cracked areas. The CZM is developed for mode-I plane strain conditions and accounts explicitly for triaxiality of the stress state by using constitutive relations. The proposed model gives more realistic results than models based on LEFM for the stiffness losses of the crack rotor system for a wide range of the crack depth. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear response of a beam under distributed moving contact load

In this paper, the nonlinear behavior of a one-dimensional model of the disc brake pad is examined. The contact normal force between the disc brake pad lining and rotor is represented by a second order polynomial of the relative displacement between the two elastic bodies. The frictional force due to the sliding motion of the rotor against the stationary pad is modeled as a distributed follower-type axial load with time-dependent terms. By Galerkin discretization, the equation governing the transverse motion of the beam model is reduced to a set of extended Duffing system with quasi-periodically modulated excitations. Retaining the first two vibration modes in the governing equations, frequency response curves are obtained by applying a two-dimensional spectral balance method. For the first time, it is predicted that nonlinearity resulting from the contact mechanics between the disc brake pad lining and rotor can lead to a possible irregular motion (chaotic vibration) of the pad in the neighborhood of simple and parametric resonance. This chaotic behavior is identified and quantitatively measured by examining the Poincaré maps, Fourier spectra, and Lyapunov exponents. It is also found that these chaotic motions emerge as a result of successive Hopf bifurcations characterized by the torus breakdown and torus doubling routes as the excitation frequency varies. Various aspects of the numerical difficulties in the solution of the nonlinear equations are also discussed.     

Vibration analysis of rotating systems with open and breathing cracks

This paper is concerned with vibration analysis of rotating systems containing cracks. The flexibility matrix of cracked element is calculated with modified integration limits which is more accurate than conventional methods. The effect of this modification on the coefficients of flexibility matrix is presented for a simple rotor system containing open crack. To model the crack breathing behavior, a new finite element approach is introduced and implemented. Then, the dynamic response of a rotor with a breathing crack is evaluated by using the frequency/time domain approach (short time Fourier transform). The ability of short time Fourier transform to detect small cracks is investigated and compared with the transient response. The results provide a possible basis for an on-line monitoring system.     

Non-stationary nonlinear analysis of a composite rotating shaft passing through critical speed

In this paper, nonlinear non-stationary dynamics of a nonlinear composite shaft passing through critical speed is studied. The nonlinearity is due to the large amplitude of shaft vibration. The equations of motion are obtained by three-dimensional constitutive relationships of composite materials. The gyroscopic effect, rotary inertia and coupling caused by material anisotropy are considered but shear deformation is neglected. Without any simplification, axial-flexural-flexural-torsional equations of motion (EOM) for the elastic composite shaft with variable rotational speed are obtained. The approximate analytical method namely asymptotic method is applied to analyze the nonstationary behavior of the composite shaft with constant acceleration. First, the EOMs are discretized using one and two-term Galerkin method. Then, the resulted equations are transformed to normal coordinates. Finally, the asymptotic method is applied to equations described in normal coordinates. Analytical expressions governing the amplitude and phase of motion during passage through critical speeds are obtained. By comparing the results obtained from analytical solutions, it is shown that discretization by one mode is not enough due to the existence of coupling in the equations and at least two modes are necessary for this purpose. Effects of damping, eccentricity, initial angular velocity and fiber angle on response amplitude are investigated. For verification, the results of perturbation theory are compared with numerical simulations and it is shown that there is good agreement between both methods.     

Dynamic analysis and experiment investigation of a cracked dual-disc bearing-rotor system based on orbit morphological characteristics

The paper is aimed to examine dynamic behaviors of a dual-disc bearing-rotor system in multi-fault state, and the crack detection based on the orbit morphological characteristics and vibration responses is proposed. Dynamic response and vibration signal analysis are two significant studies in rotor system. Most researchers have simulated the nonlinear dynamics and analyzed the fault signal using various methods separately. However, the fault feature from vibration signal is tightly connected with the dynamic mechanism in the rotor system, especially in rotor system with coupling multi-fault. In the paper, the dynamic model of the dual-disc bearing-rotor system is established, which takes into account the effects of crack, rub-impact and nonlinear oil-film forces. The vibration responses and the effect of crack on dual-disc rotor system with multi faults are investigated. The existence of crack and the coupling effect of multi faults enrich dynamic behavior of the dual-disc bearing-rotor system, and the response near the 1/2 subcritical speed provides a criterion for crack detection. Experiment investigation is attempted for the first time, which is based on the changes of crack depth and rotation speed for multi-fault dual-disc rotor system. The analysis of the dynamic response and the orbit morphological characteristics from experiment can effectively detect the crack information.     

Analysis of the effects of a pulsed electromagnetic field on the dynamic response of electrically conductive composites

The effects of pulsed electromagnetic fields on the dynamic mechanical response of electrically conductive anisotropic plates are studied. The analysis is based on the simultaneous solving of the system of nonlinear partial differential equations that include equations of motion and Maxwell’s equations. Physics-based hypotheses for electro-magneto-mechanical coupling in anisotropic composite plates and dimension reduction solution procedures for the nonlinear system of the governing equations are presented. A numerical solution procedure for the resulting two-dimensional nonlinear system of the governing equations has been developed and consists of the sequential application of time and spatial integration and quasilinearization. The developed methodology is applied to the problem of the dynamic response of a long current-carrying unidirectional carbon fiber polymer matrix composite plate subjected to transverse impact load and in-plane pulsed electromagnetic load. The interacting effects of the pulsed electric current, external magnetic field, and mechanical load are studied.     

Implementation of a cohesive zone model for crack closure analysis of rotors

The presence of a crack in a rotor introduces a local flexibility which affects its dynamic response. Moreover, the crack may open and close during the vibration period. The crack status is a function of time and also depends on the rotational speed and the vibration amplitude of the rotor. This nonlinear case is still a challenging research topic especially in the field of closing crack in the rotating shaft. A cohesive zone model is developed in order to analyze the stiffness of a crack in a rotating shaft. The proposed expression will be compared to three different crack models, namely, a breathing crack model, a switching crack model and an open crack model. Moreover, a cohesive law to predict and to analyse the stress at the crack tip is presented. The numerical model is implemented using a finite element formulation. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique

The nonlinear modeling and subsequent dynamic analysis of cracked Timoshenko beams with functionally graded material (FGM) properties is investigated for the first time using harmonic balance method followed by an iterative technique. Crack is assumed to be open throughout. During modeling, nonlinear strain–displacement relation is considered. Rotational spring model is adopted in order to model the open cracks. Energy formulations are established using Timoshenko beam theory. Nonlinear governing differential equations of motion are derived using Lagrange's equation. In order to incorporate the influence of higher order harmonics, harmonic balance method is employed. This reduces the governing differential equations into nonlinear set of algebraic equations. These equations are solved using two different iterative techniques. Methodology is computationally easier and efficient as well. This is observed that although assumption of simple harmonic motion (SHM) simplifies the problem, it yields to erroneous results at higher amplitude of motion. However, accuracy of the solution is improved considerably when the contribution of higher order harmonic terms are considered in the analysis. Results are compared with the available results, which confirm the validity of the methodology. Subsequent to that a parametric study on influence of forcing term, material indices and crack parameters on large amplitude vibration of Timoshenko beams is performed for two different boundary conditions.     

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.     

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.     

Nonlinear electro-dynamic analysis of micro-actuators: Effect of material nonlinearity

This paper presents a nonlinear dynamic analysis of a micro-actuator made of nonlinear elasticity materials. The theoretical formulations are based on Bernoulli–Euler beam theory and include the effects of mid-plane stretching due to large deformation and material nonlinearity. By employing Linstedt–Poincaré perturbation method, the nonlinear governing equation is transformed into a set of linear differential equations which are then solved using Galerkin’s method. Numerical results show that the linear constitutive relationship used in previous studies is valid for small deformation only whereas for large deformation, the nonlinear elasticity constitutive relationship must be used for accurate analysis. The effects of initial gap and beam length on the nonlinear electro-dynamic behavior of the micro-actuator are also discussed.     

Reduction Approaches for Thermogasdynamic Lubrication Problems

Gas thrust bearings are often used in low-load applications, e.g. in air cycle machines, in micro gas turbines or in rotor systems for fuel cell applications, to support a shaft in axial direction. The pressure and temperature distribution in a gas thrust bearing pad are described by the generalized Reynolds equation according to Dowson and the 3D energy equation. In this paper, two different approaches are presented in order to reduce the dimension of the governing nonlinear integro-differential equation system and in order to stabilize the solution process. In the first reduction approach, the temperature in the fluid is averaged across the fluid film according to Lee and Kim. In the second approach, Legendre polynomials are used to approximate temperature, density and fluidity across the fluid film according to Elrod, Brewe and Moraru. The reduction techniques are compared with respect to numerical efficiency, accuracy and convergence behavior. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)