Flexible Rotor-bearing System Research Articles

A Numerical Analysis for Ball End Milling Due to Coupling Effects of a Flexible Rotor-Bearing System Using GPEM

In this paper, the tool-tip responses for ball end milling, due to the coupling effects of a flexible rotor-bearing system, are investigated numerically. The milling machine tool spindle is modelled as the flexible rotor-bearing system. The critical speeds, natural modes, and unbalance responses of the system are calculated by applying the generalized polynomial expansion method. This generalized polynomial expansion method expresses the displacement as a series formed by the product of generalized coordinates and axial coordinate polynomials. According to the dynamic cutting force obtained by some scholars in the past, combined with the characteristics of the flexible rotor, the dynamic response of the tool-tip for ball end milling is numerically analyzed. The responses, including time histories, orbits, and FFT diagrams, are plotted to analyze the dynamic behaviors of the tool-tip. The coupling effects of the flexible rotor-bearing system on the system for ball end milling are first studied using the generalized polynomial expansion method. Unlike previous studies, the natural frequency varies with spindle speed and which of the different modes are included in the tool-tip response depends mainly on the spindle speed. Thanks to the gyroscopic effect, the critical speeds and responses of tool-tips can be discussed with respect to various spindle speed and tool flutes. The natural modes are accurately determined, and will excite critical speeds for certain modes, including forward and backward modes, thereby significantly affecting tool-tip response. In addition, the cutting force component associated with the tool-tip response affects the rotor-bearing system parameters, complicating the issue. Milling at higher spindle speed (2160–19,950 rpm), an important new result is found that the tool-tip oscillates with the cutting-force frequency, accompanied by a longer period vibration of the first backward mode of the rotor-bearing system. It can also be seen from the frequency spectrum analysis that, as the spindle speed increases, the peak amplitude of the first backward mode becomes larger. Milling at lower spindle speed (960, 1320 rpm), the in-plane vibration trajectory of the tool-tip gradually expands outwards clockwise around the origin until a stable loop is reached. This is because only the first backward mode of the rotor-bearing system is excited. Considering the coupling effect of the rotor-bearing system to perform the vibration analysis of the milling machine system, the parameters of the system can be designed or the spindle speed can be selected to avoid severe vibration during machining.

A vibration model of a flexible rotor-bearing system considering the defect in a double-row cylindrical roller bearing

Vibration analysis of double-row cylindrical roller bearing systems is useful for controlling the vibrations of rotating transmission systems. However, most current works only formulated the rotor system with single-row cylindrical roller bearings. Moreover, the vibration characteristics of the rotor system with defective double-row cylindrical roller bearings are not clearly. To overcome this problem, a vibration model of a flexible rotor system with the defective double-row cylindrical roller bearing is proposed in this work. The effects of the speed and defect on the vibrations of flexible rotor systems are studied. The vibrations from flexible and rigid models are compared under different rotation speeds. Note that the vibration waveforms from rigid and flexible models are similar. The vibrations of different nodes are different under different speeds, which cannot be formulated by the previous rigid models. The vibration model can better represent the vibrations of rotor systems with and without the defect. Moreover, this paper can provide a comprehensive analytical method for vibrations and acoustics of flexible rotor systems with the double-row cylindrical roller bearings.

Dynamic of a Flexible Rotor-Bearing System Supported by Worn Tilting Journal Bearings Experiencing Rub-Impact

Wear usually occurs in the loaded part of hydrodynamic bearings and leads to bearing geometry imperfections. This paper investigates the effects of wear-induced geometric imperfections of tilting pad journal bearings (TPJBs) on the dynamic behavior of the system. Furthermore, the effect of wear on the journal-bearing rub-induced contact pressure severity is investigated. A novel tribo-dynamic model is proposed for a flexible rotor-worn TPJB which integrates a mixed elastohydrodynamic model with a rotor-worn TPJB thermal and dynamic model to assess the effects of the bearing wear progression on rotor-TPJB behavior. Based on the results, wear changes the temperature distribution of the pads and oil film as well as the dynamic behavior of the system. Dynamic simulations reveal a higher vibration level and contact pressure for the worn TPJBs near the system’s critical speed and service speed. Finally, thermal and dynamic condition indicators are suggested to detect TPJB wear severity at its early stages.

Experimental parameters identification of a flexible rotor system equipped with smart magneto-rheological bearing

Magnetorheological (MR) fluid is one of the most effective smart fluids with controllable rheological and viscoelastic properties that is used in semi-active vibration control of systems and structures. MR squeeze-film dampers (MRSFDs) are categorized among smart bearings with variable stiffness and damping properties, which enable vibration mitigation of rotary systems. In this aspect, precise modeling of the rotor systems equipped with MRSFDs and identifying their parameters are of paramount importance for analyzing the vibration behavior of the system and developing control strategies. The present study aims to present the analytical tools and algorithms for identifying the parameters of flexible rotor systems equipped with MRSFDs. For this purpose, a rotor-bearing system test setup including a flexible rotor, two ball bearings, an MRSFD, and an unbalanced disk is designed and fabricated, and its governing equations of motions are derived using finite element (FE) methods. The parameters of the flexible rotor system, as well as the stiffness and damping coefficients of the bearings and nonlinear damping forces of the MRSFD, are then identified using the developed algorithm and the harmonic responses of the system as its input. The procedure is then validated by comparing the results of the FE model with identified parameters with those measured experimentally. It is shown that the developed algorithm and FE model are capable of accurately identifying the parameters of the flexible rotor-bearing system equipped with an MRSFD.

Vibration characteristic analysis of flexible rotor-bearing system subjected to external combined loads

This paper proposed a general five degrees of freedom dynamic model for investigating the vibration characteristics of flexible rotor-bearing system based on nonlinear elastic Hertz contact theory and Timoshenko beam theory by employing differential quadrature finite element and improved Newton-Raphson methods. The flexible rotor-bearing system is made up of stepped shaft and angular contact ball bearing (ACBB). First, the validation of the established model is verified by the existing literatures and finite element software ANSYS 18.2. Then, the dynamic behaviors of ACBB are discussed systematically for the investigations of the vibration behaviors of flexible rotor-bearing system provide reference. Finally, the vibration characteristics of flexible rotor-bearing system subjected to external combined loads are investigated from different perspectives thoroughly. The results show that the preload, external force, radial force and torque have important influence on the vibration behaviors of flexible rotor-bearing system.

Condition Monitoring and Identification of Misalignment with Initial Unbalance of Flexible Rotor-Bearing System

Condition Monitoring and Identification of Misalignment with Initial Unbalance of Flexible Rotor-Bearing System

Stability analysis of horizontal symmetric flexible rotor mounted on hydrodynamic bearing and squeeze film damper using analytical approach

The paper deals with the development of a close form solution to find the stability of flexible symmetric horizontal rigid and tuned flexible rotor mounted on hydrodynamic bearing with squeeze film damper system. The correlations are developed analytically to find the static equilibrium position of a rigid and flexible rotor-bearing-damper system and stability threshold of the system about its static equilibrium position under isoviscous conditions. Modified Reynolds equation with short damper approximation is used to derive the forces of hydrodynamic bearing and squeeze film damper using π-film approximation under both laminar and turbulent conditions. A simple iterative method is used to predict the stability threshold. Effects of shaft flexibility, damper support flexibility, damper mass and turbulence on stability are discussed. Deployment of squeeze film damper increases the stability limit of horizontal symmetric tuned flexible rotor-bearing system to nearly two times of its original limit and mass of damper play important role to increase the stability threshold of the system. Stability threshold of rigid system can be increased significantly by increasing the damper mass and stiffness. Insertion of turbulence reduces the limits of stability. Further work has been extended to find Thermo-hydrodynamic analysis of rotor -bearing -damper system by solving the Energy equation with adiabatic boundary conditions simultaneously with Reynolds equation with analytical approach. In case of rigid rotor bearing damper system, Thermo-hydrodynamic analysis overestimates the stability threshold as compared to isoviscous conditions, stability limit of horizontal symmetric tuned flexible rotor-bearing -damper system under estimates in comparison with isoviscous conditions till k⇀=1. Stability threshold of rigid system can be increased significantly by increasing the damper mass and stiffness.

A novel iteration method for estimation of bearings dynamic coefficients in the rotor-bearing system

A novel iteration method is designed based on a hybrid augmented Kalman filter (HAKF) algorithm and linear global-state model of the flexible rotor-bearing system to identify the dynamic coefficien...

Nonlinear Vibration Analysis of a Flexible Rotor Supported by a Journal Bearing Considering Journal Angular Motion

Abstract The effect of bearing length to diameter (L/D) ratio and large disk position on nonlinear vibration of a flexible rotor-bearing system was investigated. The rotor consisted of a shaft modeled by one-dimensional finite elements (FEs) and disks. It was supported by a self-aligning ball bearing (BB) and an axial-groove journal bearing (JB). Two JB's L/D ratios of 0.4 and 0.6, two large disk positions of 340 and 575 mm measured from the BB, and two bearing models that consider both journal's lateral and angular motion (model A) and consider only journal's lateral motion (model B) were investigated. The degrees-of-freedom (DOF) of the equation of motion (EOM) were reduced to those of the boundary DOF by real mode component mode synthesis (CMS) that retains only the first forward and backward modes of the internal DOF. Shooting method and Floquet multiplier analysis were applied to the reduced EOM to obtain limit cycles and their stability, which indicates Hopf bifurcation type. Numerical results indicated that supercritical bifurcation only occurred in the case of L/D = 0.4 and large disk position 575 mm for both bearing models. Otherwise, the subcritical bifurcation occurred except the case of L/D = 0.6 with the large disk position 575 mm that supercritical bifurcation occurred if model B was used. The experiment with the same parameters used in the calculation was conducted as verification. The experimental results showed the same bifurcation type as calculated by using model A.

On-site high-speed balancing of flexible rotor-bearing system using virtual trial unbalances at slow run

An identification algorithm is developed to execute high-speed balancing of the flexible rotor system supported on the conventional bearings, while rotating the system below its critical speed. The system is integrated with Active Magnetic Bearings (AMBs) as a suppression actuator as well as an excitation actuator. The AMB suppresses the vibration of the system and allows to introduce virtual trial unbalances to identify the orientation and magnitude of the residual unbalances present in the system. An Advanced Influence Coefficient Method (AICM) is developed that utilizes the influence coefficients obtained at high speed and unbalances identified at the low speed to effectively estimate the balance masses required for the high-speed flexible rotor balancing. The influence coefficients are required to be obtained just the once, whereas the balance masses can be estimated periodically by identifying the unbalances at low speeds. After balancing, the system can easily cross its critical speeds with less vibration. The AMB also controls abrupt change in vibration amplitude due to any uncertain faults, while operating at high speeds. Additionally, the virtual trial unbalances as a magnetic force generated through AMB reduces the mechanical effort involved in the placing of physical trial unbalances in the rotor system. This method allows on-site condition monitoring of the system periodically to reduce the impairment of high-speed machinery.

A Novel Method for the Dynamic Coefficients Identification of Journal Bearings Using Kalman Filter.

The dynamic coefficients identification of journal bearings is essential for instability analysis of rotation machinery. Aiming at the measured displacement of a single location, an improvement method associated with the Kalman filter is proposed to estimate the bearing dynamic coefficients. Firstly, a finite element model of the flexible rotor-bearing system was established and then modified by the modal test. Secondly, the model-based identification procedure was derived, in which the displacements of the shaft at bearings locations were estimated by the Kalman filter algorithm to identify the dynamic coefficients. Finally, considering the effect of the different process noise covariance, the corresponding numerical simulations were carried out to validate the preliminary accuracy. Furthermore, experimental tests were conducted to confirm the practicality, where the real stiffness and damping were comprehensively identified under the different operating conditions. The results show that the proposed method is not only highly accurate, but also stable under different measured locations. Compared with the conventional method, this study presents a more than high practicality approach to identify dynamic coefficients, including under the resonance condition. With high efficiency, it can be extended to predict the dynamic behaviour of rotor-bearing systems.

The modal analysis of a double disc flexible rotor-bearing system with isotropic stiffness and damping properties

This paper shows the analysis of the modal characteristics of a flexible rotor system having two non-identical discs mounted on it using the finite element method. It uses the Timoshenko beam elements theory including shear deformation, rotary bending effects and gyroscopic effects. The bearing is considered to be isotropic having translational stiffness and damping values constant in x and y directions. There are no cross-coupling terms. The conduct of the framework is investigated by reference to natural frequency maps and mode shape diagrams. A comparative study has been conducted between this system having isotropic bearings of stiffness and damping properties and the one in which the bearings have only stiffness keeping other parameters same. The gyroscopic effect is seen to be prominent at higher rotating speeds for both the cases which cause the splitting of frequency pairs into forward and backward whirls. The Campbell diagram is computed which helps in finding the critical speeds associated with each pair of diverging natural frequencies. The bearing system represented by stiffness and damping properties shows a comparatively better control over the critical speed range.

Identification of Speed-Dependent Active Magnetic Bearing Parameters and Rotor Balancing in High-Speed Rotor Systems

Estimating residual unbalances of a flexible rotor that is fully levitated on active magnetic bearings (AMBs) are challenging tasks due to the modeling error of AMB rotordynamic parameters. In this work, an identification algorithm has been developed for the estimation of dynamic parameters of speed-dependent AMBs and residual unbalances in a high-speed flexible rotor-bearing system. Parameters are identified during an estimation process with the help of displacement and current information at AMB locations only. For reducing the finite element model to suit the measurement availability, an improved dynamic reduction scheme has been proposed, which considers the gyroscopic matrix also in the transformation matrix. For a numerical testing of the developed identification algorithm, a multidisk flexible-shaft rotor is considered, which is fully levitated on AMBs. Speed-dependent AMB parameters have been modeled by a cubic function. Proportional–integral–derivative (PID) controllers are used to control the supply current to AMBs. Displacements and currents are generated using the finite element method of the rotor-AMB numerical model. These responses have been used in the identification algorithm for the estimation of the AMB displacement and current stiffness as well as of residual unbalances, concurrently. The algorithm with the proposed reduction scheme has shown an excellent estimation agreement in the presence of noisy responses and bias errors in rotor model parameters.

Application of active magnetic bearings for in situ flexible rotor residual balancing using a novel generalized influence coefficient method

ABSTRACTAn identification algorithm for the estimation of residual unbalances is developed for a flexible rotor-bearing system integrated with an active magnetic bearing (AMB). The main advantage of the present methodology is that it can be applied in situ and also with safe traversal of resonances during run-up. The flexible rotor-AMB system is modelled by the finite element method, considering randomly distributed residual unbalances along with disc discrete residual unbalances. In order to obtain equivalent discrete unbalances at limited disc locations for the balancing of the flexible rotor system, a generalized influence coefficient method (GICM) is proposed. Equivalent unbalances obtained at discrete balancing planes are used to check the reduction in unbalance responses. The proposed algorithm is compared with the conventional influence coefficient method (CICM) without AMB to check the accuracy and consistency of the estimated values of residual unbalances. The values estimated by both methods are found to be consistent, while the GICM has an advantage of using AMB for the safe operation of the unbalanced rotor at high speeds during measurements. In order to have a suitable discrete position of AMB (for effective estimation of residual unbalances), various rotor system models are considered with different positions of the AMB.

Nonlinear Dynamics of Flexible Rotors Supported on Journal Bearings—Part II: Numerical Bearing Model

The nonlinear stability of a flexible rotor-bearing system supported on finite length journal bearings is addressed. A perturbation method of the Reynolds lubrication equation is presented to calculate the bearing nonlinear dynamic coefficients, a treatment that is suitable to any bearing geometry. A mathematical model, nonlinear coefficient-based model, is proposed for the flexible rotor-bearing system for which the journal forces are represented through linear and nonlinear dynamic coefficients. The proposed model is then used for nonlinear stability analysis in the system. A shooting method is implemented to find the periodic solutions due to Hopf bifurcations. Monodromy matrix associated to the periodic solution is found at any operating point as a by-product of the shooting method. The eigenvalue analysis of the Monodromy matrix is then carried out to assess the bifurcation types and directions due to Hopf bifurcation in the system for speeds beyond the threshold speed of instability. Results show that models with finite coefficients have remarkably better agreement with experiments in identifying the boundary between bifurcation regions. Unbalance trajectories of the nonlinear system are shown to be capable of capturing sub- and super-harmonics which are absent in the linear model trajectories.

Nonlinear Dynamics of Flexible Rotors Supported on Journal Bearings—Part I: Analytical Bearing Model

To determine the bifurcation types in a rotor-bearing system, it is required to find higher order derivatives of the bearing forces with respect to journal velocity and position. As closed-form expressions for journal bearing force are not generally available, Hopf bifurcation studies of rotor-bearing systems have been limited to simple geometries and cavitation models. To solve this problem, an alternative nonlinear coefficient-based method for representing the bearing force is presented in this study. A flexible rotor-bearing system is presented for which bearing force is modeled with linear and nonlinear dynamic coefficients. The proposed nonlinear coefficient-based model was found to be successful in predicting the bifurcation types of the system as well as predicting the system dynamics and trajectories at spin speeds below and above the threshold speed of instability.

Large deflection model for nonlinear flexural vibration analysis of a highly flexible rotor-bearing system

Investigation of critical speeds and nonlinear free vibration analysis of a highly flexible rotor system have been studied. The rotary inertia and gyroscopic effect combined with inextensible geometric condition for pinned-guided shaft element have been taken into account to develop the governing equation of motion. The closed-form mathematical expressions have been derived for determining both linear and nonlinear natural frequencies and their behavioral patterns have been simultaneously demonstrated through time histories, FFTs and Poincare’s maps upon changing the control parameters. The nonlinear natural frequencies have been found to be higher by about 4–6% as compared to the findings obtained via linear analysis. For lower spin speed of the shaft, both linear and nonlinear forward natural frequencies have been found to be dominant. The influence of rotary inertia on natural frequencies by considering Euler and Rayleigh beam elements, separately has been reported. The system response has been observed to be either periodic or quasi-periodic depending on the spin speed. Outcomes which were not explored earlier enable significant theoretical understanding of free vibration analysis and whirling speeds of rotating system which are of great practical importance for investigating further dynamic performance.

Evaluation of nonlinear responses and bifurcation of a rotor-bearing system mounted on moving platform

The assessment of nonlinear phenomena with a focus on investigating the bifurcations and chaotic behaviour of an elastically induced flexible rotor-bearing system subjected to a harmonic ground motion has been studied. The higher-order Euler–Bernoulli deformation theorem has been adopted that governs the nonlinear dynamic characteristics of the rotating system. Nonlinear vibration analysis has been carried out to determine the critical speeds, i.e., Campbell diagram followed by demonstrating the inherent nonlinear signatures through the illustration of time history, Fourier spectrum and Poincare’s map upon varying the system design variables. The perturbation technique has been used to obtain the approximate nonlinear solutions from a set of polynomial algebraic equations under steady-state condition in various resonance cases. Further, stability analysis along with successive bifurcations of the solutions has been investigated. The present nonlinear model formulated based on Euler theory has been found to be capable enough to predict the correct value of critical speeds and dynamic responses in comparison with that of model developed based on Timoshenko theory. The present outcomes can offer immense practical importance for any application of the flexible rotor-bearing system when it performs high-speed operation.

Model-based interpolation-iteration method for bearing coefficients identification of operating flexible rotor-bearing system

The requirement of bearing coefficients identification based on flexible rotor-bearing systems has stimulated the investigation of bearing forces over the years, whereby in industry high speed balancing machines and flexible rotor-bearing test rigs can be used to measure the dynamic force coefficients of bearings. However, the actual rotor vibration at the bearing nodes cannot be measured or simply calculated by the data acquired from a single measurement station outside of the bearing. To solve this problem, a double-section interpolation-iteration method to identify the dynamic coefficients has been developed which uses previously identified coefficients to predict new vibration vectors from a finite element model and then updates the vibration vectors at the bearings to recalculate the coefficients. Numerical investigations on the estimation error, convergence, and robustness against noise were carried out, and the specific measurement process and method are presented. Simulation results show that this method has high confidence intervals in the identification of both direct stiffness and damping coefficients. Moreover, the experimental work, in which an active magnetic bearing was used to excite a flexible test rig supported by tilting pad bearings, was also carried out to validate the method, with the vibration shape predicted by the identified results matching the test data very well and the first forward mode parameters also matching the results from sine-swept identification.

Nonlinear dynamic modeling of a simple flexible rotor system subjected to time-variable base motions

Rotor systems carried in transportation system or under seismic excitations are considered to have a moving base. To study the dynamic behavior of flexible rotor systems subjected to time-variable base motions, a general model is developed based on finite element method and Lagrange’s equation. Two groups of Euler angles are defined to describe the rotation of the rotor with respect to the base and that of the base with respect to the ground. It is found that the base rotations would cause nonlinearities in the model. To verify the proposed model, a novel test rig which could simulate the base angular-movement is designed. Dynamic experiments on a flexible rotor-bearing system with base angular motions are carried out. Based upon these, numerical simulations are conducted to further study the dynamic response of the flexible rotor under harmonic angular base motions. The effects of base angular amplitude, rotating speed and base frequency on response behaviors are discussed by means of FFT, waterfall, frequency response curve and orbits of the rotor. The FFT and waterfall plots of the disk horizontal and vertical vibrations are marked with multiplications of the base frequency and sum and difference tones of the rotating frequency and the base frequency. Their amplitudes will increase remarkably when they meet the whirling frequencies of the rotor system.