Stability Of Rotor-bearing System Research Articles

Experimental investigation of dynamic behavior of rotor system with a novel double layers flexible support tilting pad bearing

Among all kinds of journal bearings, the tilting pad bearing has the characteristics of high stability. In order to further reduce the vibration and improve the stability of rotor-bearing systems, a novel double layers flexible support tilting pad bearing (DL-FSTPB) is presented in this paper. The bearing utilizes two layers of springs, which have different functions. The offset springs control the swing angle of the tilting pad to ensure the formation of a convergent oil film, and the symmetrical springs are used to change the bearing stiffness coefficient, increase the bearing elasticity, and improve the ability to accommodate misalignment. Experiments were carried out to study the vibration characteristics of the rotor system. The results were analyzed using wavelet analysis of acceleration and displacement, waterfall plots, a wavelet analysis, a run-up process and shaft center orbits areas. The results show that the DL-FSTPB exhibits excellent vibration attenuation characteristics, which can effectively reduce the amplitude and the system vibration level.

Bifurcation analysis of rotor/bearing system using third-order journal bearing stiffness and damping coefficients

Hydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor’s expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems.

Influence of contact interface friction of bolted disk joint on motion stability of rotor-bearing system

In rotating machines, bolted joints are widely employed to connect adjacent disks to make multiple parts into an integrated one. The contact interface, subjected to tangential force introduced by unbalanced forces during normal operation, influencing the rotor dynamics, especially in the presence of relative speed, is of interest for investigations. However, there are little researches on the rotor dynamics considering the contact interface friction of the bolted disk joint. Thus, a dynamic model of the rotor-bearing system is established, which considers the friction at the contact interface between disks, nonlinear oil-film force, etc. The nonlinear dynamic phenomena of the rotor system were investigated using the fourth-order Runge–Kutta method. Bifurcation diagrams, vibration waveforms, frequency spectrums, shaft orbits, and Poincare maps are used to demonstrate the diversity of the system responses. The results indicate that the rotor system exhibits a rich diversity motion states, such as periodic-1 motion, multiple periodic motion, and quasi-periodic motion. The parametric study about the effect of friction coefficient on rotor dynamics is also conducted in the present work, which shows that a larger friction coefficient will lead the system enters into an unstable motion state at high rotating speed, and make the system enters into multiple periodic bifurcations at a higher rotating speed. The corresponding results can contribute to the further understanding of the nonlinear dynamic characteristics of the bolted joint rotor-bearing system.

Parametric instability of anisotropic rotor-bearing systems with a transverse crack

The parametric excitations caused by the time-varying stiffness characteristics of breathing cracks are important sources of instability in cracked rotor systems. The anisotropy property of journal bearings may also affect the stability of rotor systems. These two factors were, however, seldom addressed simultaneously in available investigations. In this paper, the stability of an anisotropic rotor-bearing system with a transverse crack is discussed. The breathing crack is modeled as the time-varying area moments of inertia at the cracked element. The finite element model and the eight-coefficient model are employed for modeling the shaft and the journal bearings, respectively. An efficient method based on the Hill’s method is developed to compute the complex eigenvalues of the periodically time-varying system. On this basis, the stability of the system can be assessed by checking the real parts of eigenvalues. Finally, the dynamic stability of a cracked rotor-bearing system due to the anisotropy and the speed-dependent characteristics of bearings are discussed. Numerical results show that the stability of the rotor-bearing system can be significantly influenced by the two factors, and the different anisotropic patterns play different roles on the system’s stability. Besides, the speed-dependent characteristic can make the instability regions be narrowed due to the variation of the stiffness coefficients with the rotating speed.

Threshold stability curves for a nonlinear rotor-bearing system

Design parameters of fluid film bearings affect the stability of rotor-bearing systems, which can be characterized using so called threshold speed and a threshold curve (depending on a shaft speed and a chosen design parameter). The paper brings a detailed study of the shape of the threshold curve for a rigid rotor supported on a plain bearing. A novel comparison of stability threshold curves obtained employing various approaches to the hydrodynamic force modelling for all relevant bearing parameters is presented. Hydrodynamics forces are solved using four approaches: the infinitely short (IS) approximation, the infinitely short approximation extended by multiplicative correction polynomials for finite-length journal bearings (IScor), the finite differences method and the finite elements method. The influence of run-up time and acceleration is also investigated. All obtained threshold curves are compared to the analytical expressions for the threshold curves calculated for the IS approximation.

Analysis of dynamic characteristics and stability prediction of gas bearings

PurposeThe prediction model to estimate the stability of a rotor-bearing system is established, which can predict the stability of gas bearings by applying Routh–Hurwitz stability criterion. This paper aims to provide the theoretical foundation for controlling actively the bearing running stiffness and damping and stemming the instability of a gas film.Design/methodology/approachThe nonlinear dynamic lubrication analysis mathematical model of spherical hybrid gas bearings is established. Perturbation control equation is derived by the partial derivative method. The finite difference method is used to discrete the perturbation control equation in generalized coordinate system, and the difference expression of perturbation pressure is derived. The relational expression which involves the relationship between the dynamic characteristic coefficients of HSGHGB systems and perturbation pressure is deduced. So, the transient perturbation pressure distribution of a three-dimensional micro gas film, nonlinear gas film force, dynamic stiffness and dynamic damping coefficients of bearings are numerically computed using VC++6.0 programs.FindingsThe results show that the influence of supply pressure, speed and eccentricity on the dynamic characteristics of bearings is significant.Originality/valueThe influence law of supply pressure, speed and eccentricity ratio on the dynamic stiffness and damping coefficients of HSGHGB systems is researched. The prediction model to estimate the stability of rotor-bearing system is established, which can predict the stability of gas bearings by applying the Routh–Hurwitz stability criterion.

Dynamic Stability Analysis and Critical Speed of Rotor supported by a Worn Fluid film Journal Bearings

In this paper, the effect of wear in the fluid film journal bearings on the dynamic stability of rotor bearing system has been studied depending on the development of new analytical equations for motion, instability threshold speed and steady state harmonic response for rotor with offset disc supported by worn journal bearings. Finite element method had been used for modeling the rotor bearing system. The analytical model is verified by comparing its results with that obtained numerically for a rotor supported on the short bearings. The analytical and numerical results showed good agreement with about 8.5% percentage error in the value of critical speed and about 3.5% percentage error in the value of harmonic response. The results obtained show that the wear in journal bearing decrease the instability threshold speed by 2.5% for wear depth 0.02 mm and 12.5% for wear depth 0.04 mm as well as decrease critical speed by 4.2% and steady state harmonic response amplitude by 4.3% for wear depth 0.02 mm and decrease the critical speed by 7.1% and steady state harmonic response amplitude by 13.9% for wear depth 0.04 mm.
 

Numerical and Experimental Stability Investigation of a Flexible Rotor on Two Different Tilting Pad Bearing Configurations

Rotordynamic stability is crucial for high performance centrifugal compressors. In this paper, the weighted instrumental variable (WIV) based system identification method for rotating machinery stability is investigated based on a sine sweep forward excitation with an electromagnetic actuator. The traditional multiple input multiple output (MIMO) frequency response function (FRF) is transformed into a directional frequency response function (dFRF). The rational polynomial method (RPM) combined with WIV is developed to identify the rotor’s first forward mode parameters. This new approach is called the COMDYN method. Experimental work using the COMDYN method is carried out under different rotating speeds, oil inlet temperatures, and pressure conditions. Two sets of bearings with preloads 0.1 and 0.3 are investigated. A numerical rotor-bearing model is also built. The numerical results correlate reasonably well with the experimental results. The investigation results indicate that the new method satisfies the desired features of rotating machine stability identification. Furthermore, the system log decrement was improved somewhat with the increase of oil inlet temperature. The increase of oil supply pressure affects the rotor-bearing system stability very slightly. The results of this paper provide new and useful insights for potentially avoiding instability faults in centrifugal compressors.

Research on Periodic Motion Stability of Rotor-Bearing System with Dual-Unbalances

Multiple freedom degrees model of rotor-bearing system taking many factors into account is established, the Newmark-β and shooting method are combined during the stability analysis of periodic motion in such system. The paper focused on the influence law of two eccentric phase difference on the instability speed of rotor-bearing system. The results have shown that the instability speed rises constantly with the eccentric phase difference angle increasing in small eccentricity system. When the two unbalance be in opposite direction, the system reached its maximum instability speed. However, the unstable bifurcation generates mutation phenomenon for large eccentricity system with the eccentric phase difference angle increasing. In summary, the larger initial phase angle can inhibit system instability partly. The conclusions have provided a theoretical reference for vibration control and stability design of the more complex rotor-bearing system.

Dynamic stability of rotor-bearing systems subjected to random axial forces

This paper investigates the lateral vibration of a spinning disk–shaft system supported by a pair of ball bearings and subjected to a pair of random axial forces at both ends. The axial forces are assumed as the sum of a static force and a random process with a zero mean. Due to the random axial forces, the rotor-bearing system may experience parametric random instability under certain situations. In this work, the finite element method is applied to yield a set of discretized system equations first. The set of discretized system equations is partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. The stochastic averaging method is then adopted to obtain Ito's equations for the response amplitudes of the system. Finally the first- and second-moment stability criteria are utilized to determine the stability boundaries of the system. Numerical results show that the rotor-bearing system is always stable in the sense of the first-moment stability, and the effects of the average axial compressive force and the disk mass, which will lower all frequencies of the system, tend to destabilize the second-moment stability of the system.

Optimum design of rotor-bearing system stability performance comparing an evolutionary algorithm versus a conventional method

The main intent of this paper is to formulate, demonstrate, and validate a practical means of implementing an evolutionary optimization technique in a rotor-bearing system. The optimum design of a flexible rotor supported on three-lobe bearings is studied for the optimal performance considering system stability along with other design criteria such as fluid film thickness, power loss, film temperature, and film pressure using the genetic algorithm and the method of feasible directions. The results for different operating speed values obtained and presented in this study are to provide a comparison of these two methods, and to show the potential of the genetic algorithm in optimization of rotor-bearing systems. The genetic algorithm obtained reasonably good results for the objective function and comparable to the method of feasible directions. Thus, the genetic algorithm provides the designer an alternative design optimization approach for rotor-bearing systems.

An Investigation into the Physics Behind the Stabilizing Effects of Two-Phase Lubricants in Journal Bearings

Solid phase particles in two-phase lubricants used in journal bearings undergo deformation in the bearing clearance space during large amplitude journal whirl. The orbital energy causing instability of the rotor shaft is drawn from the drive and it is proportional to the orbital area whereas energy dissipation offered by the fluid film is a path function. The balance of these two energies decides the stability of rotor-bearing system. In the first part of this paper, equivalent snubber springs have been used to represent contact with solid phase particles and purely elastic collision is modeled. It is analytically shown that any perturbation of a circular orbit increases dissipative work, which may entrain the response of an otherwise unstable system (i.e., without perturbation) to a stable orbit. In the second part of this paper, a rigid rotor supported on journal bearings using two-phase lubricants is studied and it is shown that discontinuity in the support forces arising out of contact with solid-phase particles, irrespective of the nature of deformation of particles, is responsible for controlling the whirl orbit size.

Experimental identification of linear oil-film coefficients using least-mean-square method in time domain

In general sense, under small perturbation the stability of rotor-bearing system can be determined by linear oil-film coefficients of hydrodynamic bearing and oil-film forces can also be expressed by these coefficients. This paper proposes an experimental method to identify these coefficients and presents their characteristics under various operational conditions. A delicate test rig is constructed and experimental data are acquired under various testing conditions. From the experimental data, the relative velocity of the journal and the oil-film forces can be obtained by using a differentiator. The coefficients are identified using least-mean-square method in time domain. Some identified results are compared with the theoretical data. The experimental results indicate that the linear oil-film dynamic coefficients are sensitive to the excitation amplitude. The intensity of sensitivity is varied for different coefficients. From the investigation, it can be concluded that the linear oil-film force model will be invalid once the condition of small perturbation is not satisfied.

Stability analysis of a rotor bearing system due to surface waviness and number of balls

In the paper, the stability analysis of a rigid rotor supported by ball bearings has been studied. In the analytical formulation, the contacts between balls and races are considered as nonlinear springs, whose stiffnesses are obtained by using Hertzian elastic contact deformation theory. The implicit type numerical integration technique Newmark-β with Newton–Raphson method is used to solve the nonlinear differential equations iteratively. The effects of surface waviness and the varying number of balls on stability of rotor bearing system are observed. All results presented in form of fast fourier transformations show that the vibration characteristics of the rotor and its bearings change, when the bearings operate in different regions of their nonlinear load deflection characteristics. From the analysis, it is implied that the number of balls and number of waves in the ball bearing are two important governing parameters affecting its dynamic behavior.

Effect of Lubricant Non-Newtonian Behavior and Elastic Deformation on the Dynamic Performance of Finite Journal Plastic Bearings

This paper presents a numerical study of the performance of a dynamically loaded finite journal plastic bearing lubricated with a non-Newtonian fluid, taking into account the elastic deformation of the bearing. The non-Newtonian characteristics are adopted in this paper through an equivalent power-law. An expression for a modified Reynolds equation is derived in order to obtain the pressure gradient. Elastic deformation of the bearing surface was estimated in a direction normal to the bearing surface using Boussinesq equations. The film shape was modified accordingly and then iterated with the hydrodynamic pressure distribution in the bearing until a convergent solution was obtained. The Reynolds equation was solved numerically, considering three values of the flow-behavior index (n = 0.6, 1, and 1.2) and a wide range of journal speeds, materials, and clearance ratios. Consequently, the finite perturbation technique was used to determine the eight values of oil film stiffness and damping coefficients. By using the dynamic coefficients, the stability characteristics of the rotor-bearing system and the critical speed were calculated. The results show that increasing the flow-behavior index enhances the rotor-bearing system stability. A considerable destabilizing effect is obtained upon decreasing the elastic deformation coefficient.

Stability analysis of rotor-bearing systems via Routh-Hurwitz criterion

A method of analysis is developed for studying the whirl stability of rotor-bearing systems without the need to solve the governing differential-equations of motion of such systems. A mathematical model, comprised of an axially-symmetric appendage at the mid-span of a spinning shaft mounted on two dissimilar eight-coefficient bearings, is used to illustrate the method. Sufficient conditions for asymptotic stability of both the translational and rotational modes of motion of the system have been derived. The system's stability boundaries, presented graphically in terms of the various system non-dimensionalized parameters, afford a comprehensive demonstration of the effects of such parameters on the system's stability of motion.

Field Methods for Identification of Bearng Support Parameters—Part I: Identification From Transient Rotor Dynamic Response due to Impacts

A simple procedure, with the potential as a field resource, for identification of a bearing support parameter from recorded transient rotor responses due to impact loads follows. The method is applied to a test rotor supported on a pair of mechanically complex bearing supports, each comprising a tilting pad bearing in series with an integral squeeze film damper. Identification of frequency dependent bearing force coefficients is good at a rotor speed of 2000 rpm. Stiffness coefficients are best identified in the low frequency range (below 25 Hz) while damping coefficients are best identified in the vicinity of the first natural frequency (48 Hz) of the rotor bearing system. The procedure shows that using multiple-impact frequency averaged rotor responses reduces the variability in the identified parameters. The identification of frequency-dependent force coefficients at a constant rotor speed is useful to assess rotor-bearing system stability.

The Experimental Study of Aerodynamic Plate-Foil Journal Bearings for High Speed Cryogenic Turboexpander

Experiments have been conducted on the application of a new type of gas lubricated self-acting journal bearing—plate-foil bearing to a high speed cryogenic turboexpander. The 25.0 mm diameter bearing consists of a rigid housing and a flexible foil element; different structure parameters of the bearing are selected in the experiments. Experimental results indicate the foil bearing presented here offers better system stability performance and high durability, and it confirmed that the application of this type of plate-foil bearing to the cryogenic turboexpander is feasible. The experiment results, including the effect of the plate-foil bearing structure parameters on the rotor-bearing system stability, are described in this paper. Presented at the 54th Annual Meeting Las Vegas, Nevada May 23–27, 1999

FINITE ELEMENT ANALYSIS OF WHIRL SPEEDS FOR ROTOR-BEARING SYSTEMS WITH INTERNAL DAMPING

A finite element model belonging to a C0- class formulation is presented for the study of whirl speeds and stability of rotor-bearing systems. In addition to the effects of translational and rotatory inertia and the gyroscopic moments, the combined effects of transverse shear deformations and the internal viscous and hysteretic damping are also incorporated into the formulation. Results of forward and backward speed and damped stability are presented and compared with other previously published works. The good convergence and high accuracy of the present finite element model are demonstrated with numerical examples.

Liapunov stability analysis of a rotor-bearing system with a flexible attachment of appendage to shaft

This paper presents a general method of analysis for investigating the whirl stability of rotor-bearing system whose appendage is flexibly attached to the spinning shaft. Sufficient conditions of asymptotic stability involving system different parameters are derived based on Liapunov's theory. An inclusive analysis of the effect of the combined flexibilities of the elastic attachment of the appendage to the shaft and the two end bearings coupled with the other various parameters of the system on the dynamic stability is presented.