Unbalance Response Research Articles

Numerical Study on Unbalance Response of Dual-Rotor System Based on Nonlinear Bearing Characteristics of Active Magnetic Bearings

The magnetic suspended dual-rotor system (MSDS) has the advantage of a high power density. The system can be used in high-speed rotating machinery. The major purpose of this study is to predict the unbalance response of the MSDS considering the nonlinear bearing characteristics of active magnetic bearings (AMBs). Firstly, the nonlinear bearing model was established by a non-linear magnetic circuit method (NMCM). The model considers magnetic flux leakage, magnetic saturation, and working position flotation accurately. Then, the dynamic model of the system was established by using the finite element method and solved by the Newmark-β method. Finally, the effects of external load, rotational speeds, and control parameters were studied. Axial trajectory diagrams, stability zone diagrams, and waterfall diagrams were employed to analyze the dynamic behaviors of the MSDS. The results indicate that the external load, rotational speeds, and control parameters have a significant impact on the unbalance response of the system. Super harmonics of rotational frequencies and their combined frequencies may be excited by heavy load conditions. Appropriate control parameters can suppress the nonlinear phenomena. The obtained results of this research will contribute to the design and fault diagnosis of MSDSs.

Benchmark solutions for the dynamic analysis of general rotor-bearing system with arbitrary linear boundary conditions

The rotor unbalance is a major source of rotor vibrations. Rotor vibrations often produce many undesired effects like noise, wear and fatigue, etc. In this paper, we try to seek the benchmark solutions for the unbalance responses of complex rotor-bearing system. The presented approach could be seen as the extension of transfer matrix method (TMM) in some sense. For the TMM, a disk or supporting structure cut off one uniform shaft element into two and someone must use the compatibility condition between these two new elements to derive the transitive matrix. However, for the presented approach, it directly solves the governing equations of uniform shaft elements with consideration of the effects of disks and supporting structures. Thus, this analytical approach is advantageous in reducing the times of matrix multiplication between state matrices and field matrices. One only needs to calculate the inversion of 16 × 16 dynamic stiffness matrix to find the steady state response. It saves the computer memory and is easy to be programmed. In addition, this analytical approach avoids the problems of selecting optimal discretization mesh densities in the case of FEM applications. For arbitrary linear boundary condition, the benchmark solutions are always easy to be obtained. The numerical simulation is carried out and two numerical examples are given to validate the new solutions. In which the finite element method is used as the numerical approach. Simulation results show that the benchmark solutions match very well with the FEM results. The effects of anisotropy in the supporting structures on the rotor's dynamic behavior, which are observed in this work, are also in accordance with many references. This validates the benchmark solutions further.

An efficient method for transient response of rotor systems based on squeeze film damper

The rotor system operation includes a series of nonlinear factors, and the optimization of computational methods is vital to increasing industrial productivity. Here we show a method(LCM, local coefficient method) actualized by coefficients interpolation to rapidly calculate the unbalance response of the oil film-supported rotor system. A database with various coefficients is developed to approximate the time-varying oil film force of dampers. The performance difference between the LCM and the conventional method was given in various unbalanced masses and operating conditions. We perform experimental tests and numerical analysis to quantify the effectiveness and efficiency of the LCM. In addition, we elaborate on parameters that affect the accuracy and suggest the application range of the LCM for general rotor system.

A novel collaborative optimization assembly process method for multi-performance of aeroengine rotors

In the assembly process of aeroengine, the aeroengines have been rejected at pass-off test for high vibration of high-pressure rotor under the condition of meeting the geometric accuracy. Repeated disassembly and trial assembly will lead to interface damage, which will affect the safety and reliability of the engine. Therefore, a multi-objective collaborative optimization process method for geometric accuracy and dynamic performance is put forward based on the practical engineering problems. In the proposed method, the primal problem firstly reveals the transfer mechanism of coaxiality and unbalance of rotor system, and the prediction model of the coaxiality and unbalance of the rotor system was established. Then, combined with the high-precision reduced-order model of aeroengine high-pressure combined rotor, the system vibration responses under different assembly states are obtained. Finally, a multi-objective collaborative optimization assembly process method considering coaxiality, unbalance and vibration response is obtained. The results show that the collaborative optimization process method considering assembly accuracy and dynamic performance is the optimal assembly strategy, which can reduce the vibration amplitude of the key nodes of the rotor system while ensuring high assembly accuracy and achieve the low-level vibration of the rotor system.

Rotordynamic analysis of a complex high-speed rotor

High-speed turbomachinery such as cryogenic turboexpander, turbocharger, turbo-compressor, and dental drill machines are prone to rigorous vibration and instability. The rotors of such high-speed turbomachinery cannot be analyzed by the fundamental Jeffcott rotor model. This is because, under the conditions of high-speed operation, the flexible rotor experiences many factors such as gyroscopic effect, shear deformation, axial torque, internal friction and viscous damping. These factors can influence catastrophic subsynchronous whirl in the system. Turbomachinery rotors have complex geometries and involve discs, turbine, compressor and bearings. In addition, complex turbomachinery rotors operate in high-speed conditions where tangential forces come in to picture, which further influences the behavior of a rotor and tends to destabilize the turbomachinery. These instabilities can be addressed by the design of a suitable rotor and its supporting bearings. In the present work, a complex turboexpander rotor is designed, which is supported on gas foil bearings, and its modal behavior is studied under high-speed environment. The rotor is small-sized and incorporates a disc, a turbine, a compressor and a pair of gas foil bearings with support structures. Finite element modeling is carried out, and the bearing dynamic coefficients are considered while modeling. The dynamics of the rotor is studied by estimating critical speeds, mode shapes and unbalance response.

Modal balancing for ball bearing-rotor system using nonlinear normal modes

Rotor dynamic balancing procedure relies on the linearisation hypothesis of the rotor system. However, some mechanical elements constituting a rotor system behave in various nonlinearities, such as the contact force and radial internal clearance between rolling balls and races for a rotor-bearing system. To balance a nonlinear rotor system, the traditional modal balancing method must be extended. This paper investigates the modal balancing method for a rotor-bearing system based on nonlinear normal modes (NNMs). The proposed algorithm includes three main steps. First, compute the NNMs and express nonlinear modal shapes using complex Fourier series. Second, describe the unbalance response as a complex amplitude containing higher harmonics. Third, estimate the correction coefficient of each harmonic, then compute the unbalance correction of each mode of interest until the total unbalance corrections are obtained. Hertz contact force and two different sizes of radial internal clearances of the ball bearings are considered during the balancing simulation. The responses after balancing with NNMs are present and compared with the linear modal method to demonstrate the accuracy and effectiveness of the proposed method.

Nonlinear Dynamics of Turbine Generator Shaft Trains: Evaluation of Bifurcation Sets Applying Numerical Continuation

Abstract The nonlinear dynamics of turbine generator shaft trains for power generation are investigated in this paper. Realistic models of rotors, pedestals, and nonlinear bearings of partial arc and lemon bore configuration are implemented to compose a nonlinear set of differential equations for autonomous (balanced) and nonautonomous (unbalanced—per ISO) cases. The solution branches of the dynamic system are evaluated with the pseudo-arc length continuation programed by the authors, and the respective limit cycles are evaluated by an orthogonal collocation method and investigated on their stability properties and quality of motion for the respective key design parameters for the rotor dynamic design of such systems, namely, bearing profile and respective pad length, preload and offset, pedestal stiffness and elevation (misalignment), and rotor slenderness. Model order reduction is applied to the finite element rotor model and the reduced system is validated in terms of unbalanced response and stability characteristics. The main conclusion of the current investigation is that the system has the potential to develop instabilities at rotating speeds lower than the threshold speed of instability (evaluated by the linear approach) for specific unbalance magnitude and design properties. Unbalance response (with stable and unstable branches) is evaluated in severely reduced time compared to this applying time integration methods, enabling nonlinear rotor dynamic design of such systems as a standard procedure, and revealing the complete potential of motions (not only local).

A Synchronous Holo-Balancing Method for Flexible Rotors Based on the Modified Initial Phase Vector

Reducing the vibration responses caused by the unbalance of the rotor is a crucial technology to ensure the safe and efficient operation of high-speed rotating machinery. For the flexible rotor system, the unbalance responses of rotors vary in different radial directions due to the anisotropic stiffness. The holospectrum technique using multi-sensor fusion provides ideas to solve this issue. The standard balancing method based on the holospectrum technique takes the Initial Phase Vector (IPV) as the balancing object, which could not represent the unbalance of flexible rotors correctly. In addition, the operating speed of the flexible rotor is generally higher than the first critical speed (FCS), making it necessary to fulfill the balancing of the first two modal shapes respectively in two steps. The above-mentioned procedure is not only very dangerous but also requires a large number of test runs resulting in corresponding wastages. In order to solve the issues mentioned above, a synchronous holo-balancing method for flexible rotors based on the modified initial phase vector (MIPV) is developed in this paper. Firstly, to tackle the limitation of IPV, the concept of equivalent rotating frequency circle is presented and then the MIPV is proposed as a new balancing object. MIPV could linearly represent the unbalance of the flexible rotor and increase the accuracy of holo-balancing. Secondly, the synchronous holo-balancing method is carried out to achieve the balancing at the speed below the FCS, reducing the complexity and danger of test runs at high operating speeds. And the synchronous balancing procedures based on the MIPV are further summarized in detail. Finally, two experimental results with different operating speeds and probe installations validate the feasibility and effectiveness of the proposed method. Compared with other traditional balancing methods, the proposed method can minimize the residual vibration and achieve state-of-the-art performances.

Interval model validation for rotor support system using Kmeans Bayesian method

To identify the uncertain stiffness and damping of the rotor support system, and obtain the most credible interval parameters, an interval model validation for rotor support system using Kmeans Bayesian method is proposed. Interval model updating is transformed into interval lower bound and interval diameter determination. The minimization of discrepancies between predicted and experimental data is converted into minimizing the area between the cumulative distribution function (CDF) of the errors and CDF of 0. To improve the computational efficiency, the long short-term memory (LSTM) network is constructed to denote the mapping relationship between the unbalance response and support parameters. To obtain the most credible interval, the Kmeans clustering algorithm is combined with the Bayesian method to select the interval bounds with maximum credibility. Kmeans clustering algorithm is used to reduce the number of reconstructed errors while utilizing all the information on reconstruction errors. Bayesian method is used to calculate the credibility of the interval bounds. The performance of the proposed method is validated by a gas-generator rotor and a dual-disks rotor. Results show that the proposed method is effective for model validation of the rotor support parameters.

Experimental investigation for the dynamic mechanical properties of a vertical anisotropically supported rotor

The dynamic mechanical properties of rotors with anisotropic supporting stiffness have been intensively studied in recent decades. However, the transient unbalance response and the whirl motion when passing through backward critical speeds have not been adequately performed. In addition, there is little published experimental data on the influence of the support stiffness anisotropy. This paper attempts to investigate the steady-state and transient unbalance response of the rotor with anisotropic supports. Response characteristics are summarized based on a modified Jeffcott model first. Next, A vertical Jeffcott rotor test rig with anisotropic elastic supports is established, and the stiffness property of different supports is measured. The steady-state unbalance response and the transient response when accelerating are measured accurately and investigated thoroughly. The influence of stiffness anisotropy and the sensor mounting inclination are investigated experimentally, and whirl modes are observed and explained. The results reveal that due to the stiffness anisotropy in the supports, the amplitude and the number of peaks of the steady-state unbalance response in the x and y directions are different, and backward whirl motions can be aroused both in the steady-state and transient unbalance response. The diversity of critical speeds and range of backward whirl regions reduce for the decreasing of stiffness anisotropy. Then we found that the number of peaks and the amplitude of transient unbalance response are relevant to the sensor mounting inclination. Moreover, the experimental results confirm the appearance of straight-line orbits and the alternations of the whirl modes when passing through critical speeds, and the mechanism are specified using the beat vibration theory as well.

Investigation of stiffness and damping coefficients in fluid film bearing with bio-oils and mineral-based oil

Rotating machinery utilizes fluid film bearings because of their good load carrying capacity and inherent damping properties. Fluid film bearings offer stiffness and damping parameters to ensure dynamic stability. The damping and stiffness characteristics of oil film significantly change the critical speed and unbalance response of the rotor. The damping and stiffness coefficients (direct, cross-coupled) may lead to system instability, also known as self-excited vibrations. Bio-oils have good tribological and physiochemical properties. They are also environment-friendly and bio-degradable. In this work, bio-oils (rapeseed, palm olein, and soybean) are evaluated in terms of their dynamic characteristics to that of mineral-based SAE40 lubricant for journal-bearing applications. The analytical extraction of damping and stiffness coefficients is performed based on the experimental dynamic viscosity data at different temperatures. In comparison to SAE40, the bio-oils have shown improved responsiveness. Bio-oils showed lower cross-coupled stiffness coefficients than SAE40, the primary source of rotor instability. The main contribution of this study is that it provides damping and stiffness coefficients for various bio-oils at different eccentricity ratios and viscosities for different types of oils. This may be used to characterize the dynamic properties of the fluid film bearing under various operating circumstances. It is concluded that the difference in cross-coupled stiffness favors bio-oils. Whereas, the difference in cross-coupled damping favors SAE40 by around 6%. This discrepancy is decreased to less than 2% for all oil’s final eccentricity ratio values (bio-oils and SAE40).

Speed-Dependent Bearing Models for Dynamic Simulations of Vertical Rotors

Many dynamic simulations of a rotor with a journal bearing employ non-linear fluid-film lubrication models and calculate the bearing coefficients at each time step. However, calculating such a simulation is tedious and computationally expensive. This paper presents a simplified dynamic simulation model of a vertical rotor with tilting pad journal bearings under constant and variable (transient) rotor spin speed. The dynamics of a four-shoes tilting pad journal bearing are predefined using polynomial equations prior to the unbalance response simulations of the rotor-bearing system. The Navier–Stokes lubrication model is solved numerically, with the bearing coefficients calculated for six different rotor speeds and nine different eccentricity amplitudes. Using a MATLAB inbuilt function (poly53), the stiffness and damping coefficients are fitted by a two-dimensional polynomial regression and the model is qualitatively evaluated for goodness-of-fit. The percentage relative error (RMSE%) is less than 10%, and the adjusted R-square (Radj2) is greater than 0.99. Prior to the unbalance response simulations, the bearing parameters are defined as a function of rotor speed and journal location. The simulation models are validated with an experiment based on the displacements of the rotor and the forces acting on the bearings. Similar patterns have been observed for both simulated and measured orbits and forces. The resultant response amplitudes increase with the rotor speed and unbalanced magnitude. Both simulation and experimental results follow a similar trend, and the amplitudes agree with slight deviations. The frequency content of the responses from the simulations is similar to those from the experiments. Amplitude peaks, which are associated with the unbalance force (1 × Ω) and the number of pads (3 × Ω and 5 × Ω), appeared in the responses from both simulations and experiments. Furthermore, the suggested simulation model is found to be at least three times faster than a classical simulation procedure that used FEM to solve the Reynolds equation at each time step.

Unbalance Response Analysis of a Spindle Supported on Gas Bearings: A Comparison between Different Approaches

Gas journal bearings are widely employed in high-speed spindles for the micromachining industry. Compared to their oil and rolling counterparts, gas bearings have a longer life span, lower friction and a lower level of noise. In order to design accurate high-speed spindles supported by externally pressurized gas bearings, it is vital to analyze the characteristics of rotor bearing systems. In this paper, we present an analysis of the unbalance response of a high-speed spindle supported by gas journal bearings. A number of aspects to enhance the accuracy of the system are discussed. We performed the analysis by considering a nonlinear and a linearized numerical model validated through experimental measurements.

Theoretical and Numerical Studies on Simultaneous Identification of Rotor Unbalance and Sixteen Dynamic Coefficients of Two Bearings Considering Unbalance Responses

Theoretical and Numerical Studies on Simultaneous Identification of Rotor Unbalance and Sixteen Dynamic Coefficients of Two Bearings Considering Unbalance Responses

Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotor: A Theoretical and Numerical Investigation on the Continuous Model and Analytical Solution for Unbalance Responses

Continuous rotor dynamics remains stagnant. In this paper, aim at multi-span and multi-disc rotor-bearing system, the continuous rotor dynamic analysis method (CRDAM) is proposed. The force acting on the shaft by the rotating eccentric disc is simulated as a point force. The counterforce of bearing is also considered as a point force. The shaft is considered free-ended. A continuous rotor dynamic model is obtained and an analytical solution is proposed to express the unbalance response as function of the position, unbalance, support stiffness and damping. The proposed method is validated by numerical experiments in which unbalance responses obtained by it are compared with that obtained by the two classical methods the finite element method (FEM) and Ricatti method. The results indicate that the proposed method is applicable to calculating unbalance response of multi-disc and multi-span rotor. Moreover, it is closer to FEM than Ricatti and can be applied to actual high speed rotors. Among the three methods, the calculating speed of Ricatti is the fastest, CRDAM is the second fastest and FEM is the slowest. The proposed method, which solves the forward problems of the continuous rotor dynamics for the multi-disc and multi-span rotors, can provide theoretical basis for further studies on inverse problems such as identification of rotor unbalance and bearing stiffness and damping coefficients without test runs and external excitations.

Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotor: A Theoretical and Numerical Investigation on the Identification of Bearing Coefficients from Unbalance Responses

Identification of bearings’ stiffness and damping coefficients, which strongly affects the dynamic characteristics of rotors, is another inverse problem of Rotor Dynamics. In this paper, aiming at multi-disc and multi-span rotors, two novel algorithms are proposed for identifying each bearing’s coefficients based on the continuous rotor dynamic analysis method. A linear functional relationship between the main complex coefficients and the cross-coupled complex coefficients is obtained, which eliminates the coupling between the coefficients and the rotor unbalance in the forward problem. Then, Algorithm I is proposed. However, it is only suitable for rolling-bearing. To solve the problem, changing the rotating speed slightly is proposed to solve the difficulty that another set of equations cannot be developed because the slope of the proposed linear function is constant when the rotating speed is maintained at a fixed speed. Then, Algorithm II, which can be applied to both rolling-bearing and oil-journal bearing, is provided. Numerical investigations are conducted to study the two methods. It is indicated that there should be a measuring point, called an adjustment point, near each bearing, whose coefficients should be identified, to obtain high identification accuracy. Moreover, the identification accuracy of the two algorithms is strongly related to sensor resolution. When the measuring errors of all the required unbalance responses are zero or the same, the identification errors are almost equal to zero. In conclusion, the proposed algorithms provide a method for monitoring the stiffness and damping coefficients of all bearings in a multi-disc and multi-span rotor under operation conditions to predict rotor dynamic behavior for the safe and steady running of rotating machines.

Continuous Rotor Dynamics of Multi-Disc and Multi-Span Rotors: A Theoretical and Numerical Investigation of the Identification of Rotor Unbalance from Unbalance Responses

Rotor unbalance identification plays a critical role in balancing rotors. In this paper, concerned with multi-disc and multi-span rotor-bearing systems, two novel algorithms called the Single Direction Algorithm (SDA) and the Two Orthogonal Direction Algorithm (TODA) are proposed for identifying rotor unbalance from unbalance responses. A matrix method is proposed to solve the problem of the equations being non-linear transcendental, there being too many unknown variables in the equations, and rotor unbalances and bearing coefficients being coupled together. The unbalance responses at all the eccentric discs are necessary for identifying their unbalances. Numerical simulations are conducted to validate the proposed methods. Moreover, an adjustment point is found, and a proper sensor resolution is suggested to achieve high identification accuracy by means of numerical studies. In addition, the identification accuracy of SDA is better than TODA, and SDA is more practical and suitable for medium-speed and high-speed rotors. The proposed algorithms have the flexibility to incorporate any number of bearings and discs and provide a technique for monitoring rotor unbalance without test runs or external exciters.

Bearing Seat Vibration Modes Undergoing Unbalanced Excitation of Multirotating Drums

Transmission modes of multiple rotating parts on combine harvester are complex and diverse, which resulted in large vibration and poor stability when the entire machine is harvesting. Aiming at the complex vibration problem of the combine harvester threshing system, this paper established the dynamic response model of the multidrum parallel system under different transmission modes and solved the vibration characteristics of the system. An experiment on the axial unbalance response of the parallel drum system under different transmission modes was carried out. The results show that the internal units of the threshing system form a whole through the transmission system, which causes the unbalanced response of the system to be superimposed on parallel threshing drums, thereby increasing the vibration amplitude. In addition, the change of the transmission mode will cause the vibration transmission path in the system to change greatly, and the boundary conditions of the system will be changed at the same time, which will eventually lead to the change of the unbalanced response characteristics.

Dynamics Investigation on Axial-Groove Gas Bearing-Rotor System with Rod-Fastened Structure

This research report discusses the dynamic behaviors of an axial-groove gas bearings-rotor system with rod-fastened structure. The time-based dependency-compressible Reynolds equation in the gas bearing nonlinear system is solved by the differential transformation method, and the continuous gas film forces of a three-axial-groove gas bearing are obtained. A dynamic mathematical model of the rotor system with rod-fastened structure supported in two- and three-axial-groove gas bearings with eight degrees of freedom is established. The dynamic motion equation of the rod-fastened rotor system is solved by the modified Newmark-β method based on disturbance compensation, which can reduce the computing error and improve computing stability. The dynamic characteristics of the rod-fastened rotor-gas bearing system are analyzed efficiently by the diversiform unbalance responses. The influence of the position angle of the pad on the nonlinear characteristics of the rod-fastened rotor system is also studied.

Numerical Evaluation and High-Speed Rotating Test on Circular Arc Spring Dampers for Centrifugal Compressors

Abstract A circular arc spring damper (CASD) is a recently proposed fluid-film damper that has two or more arc-shaped centering springs and dual radial clearances formed by wire electric discharge machining (WEDM). CASD requires less space and weight than a conventional cage-centered squeeze-film damper (SFD). It provides linear stiffness and stable damping force in rotor-bearing systems to attenuate vibration due to imbalance or to improve rotordynamic stability. The authors have been investigated the dynamic characteristics of CASD in component-level experiments. However, their performance and applicability to real machines have not been confirmed in system-level experiments. Additionally, a theoretical means of evaluation for CASD should be established to predict its dynamic coefficients and to understand the mechanism of dynamic force generation. In the first part of this study, a numerical evaluation method using two-way fluid–structure interaction (FSI) analysis and its theoretical background is presented. Transient structural analysis and fluid-film flow analysis with a simple homogeneous cavitation model are coupled in the commercial multiphysics platform ansys. The accuracy of the method was validated by comparing the damping and added-mass coefficients with results from previous experiments. Furthermore, several aspects of the force generation mechanism and the difference from conventional SFD were studied numerically. The second part of the study addresses the application of CASD in a multistage centrifugal compressor. A combined 4-in. diameter, five-pad tilting pad journal bearing (TPJB) with four-arc type CASD was newly designed and manufactured. To prove the applicability of the developed damper bearing, a series of rotating tests were conducted at a high-speed balancing facility with a full-scale dummy rotor with a critical speed ratio (CSR) of approximately 3.1. The measured unbalance response showed a much lower amplification factor (AF) than that of the conventional TPJB without the damper, which infers a significant improvement in the stability. The measured responses agreed with the rotordynamic analysis, which uses the dynamic coefficients of CASD derived from the proposed numerical evaluation method.