Rotordynamic Behavior Research Articles

Flexural–flexural–extensional–torsional vibration analysis of composite spinning shafts with geometrical nonlinearity

In this paper, nonlinear dynamics of an unbalanced composite spinning shaft are studied. Extensional–flexural–flexural–torsional equations of motion are derived via utilizing the three-dimensional constitutive relations of the material and Hamilton’s principle. The gyroscopic effects, rotary inertia and coupling due to material anisotropy are included, while the shear deformation is neglected. To analyze the rotor dynamic behavior, the full form of the equations without any simplification assumption (e.g., stretching or shortening assumption) is used. The method of multiple scales is applied to the discretized equations. An analytical expression as a function of the system parameters describing the forced vibration of a spinning composite shaft in the neighborhood of the primary resonance is obtained. The discretization is done with both one and two modes, and the results are compared. It is shown that although the excitation is tuned in the neighborhood of the first mode, one-mode discretization is not sufficient and it leads to inaccurate results. It shows the necessity of employing at least two modes in discretization due to the coupling in the equations. The effects of the external damping, eccentricity and the lamination angle on the vibration amplitude are investigated. In addition, the effect of the extensional–torsional coupling on the frequency response curves is investigated. To validate the perturbation results, numerical simulation is used.

Simulation of the dynamic behavior of a rotor subject to base motion under variable rotational speed

Nowadays, rotating machines tend to operate at high speeds and sometimes, in complex situations such as on moving support. Thus, accurate prediction of their dynamic behavior must be made to ensure a reliable design. In this paper a finite element model is developed, to simulate the dynamic behavior of a flexible rotor with constant/variable spin speed, supported by a rigid base undergoing a general movement. Simulation results are presented and discussed for different cases of base movements and running speed variations.

Dynamic conditions to destabilize persistent Rotor/touchdown bearing contact in AMB systems

It is now common practice to supplement a magnetic bearing with a touchdown bearing to protect the rotor and stator components. Rotor/touchdown bearing contact may arise from rotor drop, caused by power loss or emergency shutdown. This paper considers the control options that are viable when the magnetic bearing is still functional should contact arise from intermittent faults or overload conditions. The problem is that bi-stable rotor responses are possible, with and without contact. If rotor contact should become persistent, the desirable course of action is to destabilize the rotor response and induce a return to contact-free levitation. In order to achieve this, it is appropriate to gain an understanding of the rotor dynamic behavior. This is determined from analytical and simulated results to reveal suitable control actions. These may be applied through the magnetic bearing control system, or by activating the touchdown bearing through a separate control loop. The issue is that standard control action for a contact-free rotor state will not be appropriate for a rotor in persistent contact since the basic plants to be controlled are significantly different. The required control action should be activated only when contact is detected. The results demonstrate that appropriately phased synchronous forcing could destabilize synchronous forward rub responses. Alternatively, small whirl motions of a touchdown bearing could also be beneficial without disturbing the main magnetic bearing control loop.

Influence of the Radial Clearance of a Squeeze Film Damper on the Vibratory Behavior of a Single Spool Gas Turbine Designed for Unmanned Aerial Vehicle Applications

This study presents a numerical investigation using the finite element method on the vibratory behavior of a single spool gas turbine designed for unmanned aerial vehicle applications. The shaft of the rotor-bearing system is supported on a front bearing composed of a deep groove ball bearing with a vibration absorber element and a rear squeeze film damper bearing. Three radial clearances for the squeeze film damper were analyzed to determine the best geometric configuration for the rear bearing, considering the rotordynamic performance of the entire system. Whirl speeds and unbalanced system responses were carefully evaluated to determine the best radial clearance for the squeeze film damper. After defining the best radial clearance, a transient analysis was performed to simulate the transition of the system through resonance, and a spectral map is presented to illustrate the vibratory behavior of the system considering the influence of all related important frequencies. The rotordynamic behavior of the system is predicted, and vibration problems are avoided. Its mechanical drawings were released to manufacturing, and the first prototype is in the experimental test phase, thus indicating that the numerical results presented in this study are consistent.

EXPERIMENTAL ANALYSIS OF HYDRODYNAMIC SIMPLE JOURNAL BEARING

The new machine design applications require high operating speed, higher power density, small size, high load carrying capacity and to full fill this requirement it is necessary to consider transmission section. So one of the most important element to be considered for design is bearing. Bearing not only support rotor weight and operating speed but also influence on rotor dynamic behaviour This constant trend towards higher power densities in rotating machinery calls for better mechanical components capable of carrying higher loads while being the same or smaller in size without sacrificing machine safety. For hydrodynamic bearings it is important that minimum film thickness never drops below a safety limit. An oil film becomes thinner if load increases. This also results in elevated temperatures, which reduce oil viscosity and further decrease oil film thickness. When the fluid-film bearings operate under high speed, heat is generated within the oil film due to shearing of the lubricant and temperature rise of the lubricant fluid film and the bearing surface takes place

Nonlinear Dynamic Analysis of a Rigid Rotor Supported By a Spiral-Grooved Opposed-Hemisphere Gas Bearing

ABSTRACTThe nonlinear dynamic behavior of a rigid rotor supported by a spiral-grooved opposed-hemisphere gas bearing is investigated in this article, focusing particular attention on its whirl motion. The finite element method combined with the finite difference method is employed to solve the time-dependent Reynolds equation that is coupled with the rotor motion considering five degrees of freedom. The rotor responses to the initial disturbance and synchronous and nonsynchronous excitations are investigated. To analyze the complicated dynamic behavior of the rotor–bearing system, the trajectories of the rotor centerline, time responses, phase portraits, power spectra, Poincare maps, and bifurcation diagrams are obtained from the numerical procedure. The results show that the conical whirl instability appears earlier than the cylindrical whirl instability with increasing rotational speed for the rotor–bearing system with no unbalance mass. Moreover, it reveals that the complex dynamic behavior of the system excited by unbalance mass varies with rotational speed and rotor mass. In addition, bifurcation diagrams employing the rotating speed and rotor mass as bifurcation parameters are obtained. Finally, the nonsynchronous excitation responses are presented, which behave in a different way than the synchronous excitation responses. The results of this study offer a further understanding of the nonlinear characteristics of spiral-grooved opposed-hemisphere gas bearings.

Rotor dynamics behavior of turbo-expander involving droplet impact

ABSTRACTThis paper dedicates on the rotor dynamics behaviour research on the turbo-expander rotor system involving droplet impact. A stochastic model based on Beta distribution and Bernoulli distribution of droplet generation is established and the formulations of droplet impact forces are deduced, which is applied on the rotor dynamics equations of the tilting pad bearing supported turbo-expander considering the temperature gradient for a further analysis. A time domain research is carried out and a conclusion that continuously droplet impact will perturb the steady vibration of the turbo-expander rotor system is obtained. Monte Carlo method is implemented for a statistics dynamics research and the results suggests that in the design of expander impellers, in order to decrease the uncertainty brought by droplet impact, the number of channels should be as few as possible, the droplet impact should be controlled to occur uniformly, and the collision, entrainment of the primary droplets and the stripping of the liquid film on the blade should be strictly restrained.

Subharmonic Resonance Due to Gap Between Geometric and Magnetic Centers of Rotor Supported by Superconducting Magnetic Bearing

Superconducting magnetic bearings (SMBs) have various merits because of stable levitation without physical contact. However, the system's low damping causes vibration with a large amplitude near the critical rotational speed, and the nonlinearity of the electromagnetic force causes complicated vibrations. Moreover, if magnetization distribution of a rotor is not uniform, the rotor's dynamical behavior can be more complicated because a superconducting bulk in a superconductive state traps the nonuniform magnetic field, which has significant influence on restoring force acting on the rotor. Therefore, it is necessary to consider influence of magnetic unbalance of the rotor on the whirling of a rotor supported by a SMB. This paper deals with nonlinear dynamics of a rotor, which has a gap between the geometric and magnetic centers of the rotor and which is supported by a SMB. We derived the equations of motion of the rotor and discussed whether subharmonic resonances of order 1/2 and order 1/3 occur by considering nonlinear terms of the electromagnetic force. By numerical calculation based on the Rung-Kutta method, we clarified that subharmonic resonances of order 1/2 can occur without disturbance, whereas subharmonic resonances of order 1/3 can occur only with disturbance. Moreover, we performed nonlinear analysis by means of the harmonic balance method. We also carried out experiments, which verified the analytical and numerical predictions of occurrence of subharmonic resonances.

Dynamics of multilayer, multidisc viscoelastic rotor – An operator based higher order classical model

Inherent material damping plays a very significant role on dynamic behaviour of rotors. The material damping in a spinning rotor produces a tangential force along the whirl direction and its magnitude being proportional to spin speed. After certain value of spin speed, decided by the characteristic of the system, the tangential force becomes strong enough to throw the rotor centre out of the whirl orbit by inflating it progressively. This leads to destabilization of the system and corresponding speed is known as stability limit of spin speed. Stability limit of spin speed for Jeffcott rotor, by using viscous form of material damping model is straight forward and has been reported by several researchers, however the same analysis for viscoelastic material characteristics is not reported much. This analysis is very relevant for industrial requirements to replace bulky and heavy metal rotor by light but strong rotors. This is achieved either by reinforcing fibre or multi layering arrangements. Both of which are represented by viscoelastic constitutive behaviour. This paper gives mathematical derivation of equations of motion for multi-disc, multi-layered rotor–shaft-system. Both lumped mass and discretized approach (finite element) are presented here mathematically and numerical simulation results are compared. The lumped mass approach gives a concise yet acceptable accuracy of the results.

Rotordynamic Analysis of a Shaft Using Magnetorheological and Nanomagnetorheological Fluid Journal Bearings

ABSTRACTThe rotordynamic behavior of a system supported by journal bearings is critical to its reliability. A suitable method of control of the orbital motion of a shaft in a journal bearing is the use of smart lubricants, in effect fluids with controllable physical properties. There are various categories of smart lubricants. One class of smart lubricants, magnetorheological fluids, are produced as a dispersion of magnetic particles in a carrier fluid, which is usually a conventional lubricant. These particles form chains under the influence of a magnetic field, which hinder the lubricant flow, thus changing its apparent viscosity. Magnetorheological fluids (MRFs) exhibit high yield stress, low delay of response, and relatively low friction while not in their active state. A subcategory of MRFs, nanomagnetorheological fluids (NMRFs) with particle size on the nanometer scale, exhibits lower yield stress than MRFs but display high viscosity. The effect of the MRFs and NMRFs on the rotordynamic behavior of a shaft is calculated through a combined finite element and computational fluid dynamics analysis. Though the MRF with the specific geometrical configuration of the bearing is not sufficiently activated and therefore does not improve the performance of the magnetorheological journal bearing, the NMRF has the ability of limiting up to 82% the amplitude of the vibrations of the shaft.

Experimental Results and Computational Fluid Dynamics Simulations of Labyrinth and Pocket Damper Seals for Wet Gas Compression

The most recent development in centrifugal compressor technology is toward wet gas operating conditions. This means the centrifugal compressor has to manage a liquid phase which is varying between 0% and 3% liquid volume fraction (LVF) according to the most widely agreed definition. The centrifugal compressor operation is challenged by the liquid presence with respect to all the main aspects (e.g., thermodynamics, material selection, thrust load) and especially from a rotordynamic viewpoint. The main test results of a centrifugal compressor tested in a special wet gas loop (Bertoneri et al., 2014, “Development of Test Stand for Measuring Aerodynamic, Erosion, and Rotordynamic Performance of a Centrifugal Compressor Under Wet Gas Conditions,” ASME Paper No. GT2014-25349) show that wet gas compression (without an upstream separation) is a viable technology. In wet gas conditions, the rotordynamic behavior could be impacted by the liquid presence both from a critical speed viewpoint and stability-wise. Moreover, the major rotordynamic results from the previously mentioned test campaign (Vannini et al., 2014, “Centrifugal Compressor Rotordynamics in Wet Gas Conditions,” 43rd Turbomachinery Symposium, Houston) show that both vibrations when crossing the rotor first critical speed and stability (tested through a magnetic exciter) are not critically affected by the liquid phase. Additionally, it was found that the liquid may affect the vibration behavior by partially flooding the internal annular seals and causing a sort of forced excitation phenomenon. In order to better understand the wet gas test outcomes, the authors performed an extensive computational fluid dynamics (CFD) analysis simulating all the different types of balance piston annular seals used (namely, a tooth on stator (TOS) labyrinth seal and a pocket damper seal (PDS)). They were simulated in both steady-state and transient conditions and finally compared in terms of liquid management capability. CFD simulation after a proper tuning (especially in terms of LVF level) showed interesting results which are mostly consistent with the experimental outcome. The results also provide a physical explanation of the behavior of both seals, which was observed during testing.

Rotordynamics of Two‐Pole Turbo Generators with Refined Modelling of the Unbalanced Magnetic Pull

AbstractWithin this contribution the rotordynamic behaviour of two‐pole turbo generators is discussed. In the investigation a basic model originally proposed by Kellenberger [1], introducing a refined description of the unbalanced magnetic pull (UMP), is reviewed and extended. The analysis comprises the investigation of lateral stability behaviour and stationary torque characteristic for a magnetic eccentricity. There are two main results of the contribution: firstly it is shown, that the non‐radial component of the UMP has the same structure as the restoring forces due to a non‐circular shaft, but opposite sign. Secondly it can be seen from the extended model, that an eccentric point of application of the UMP may change the systems stationary torque characteristic significantly. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear Analysis of Rub Impact in a Three-Disk Rotor and Correction Via Bearing and Lubricant Adjustment

Rubbing between rotating and stationary surfaces in turbomachinery can result in catastrophic failures if not caught quickly. Removing the rub impact can then often require time consuming and expensive solutions including field balancing or magnetic bearing systems. However, simple changes in bearing dynamics via bearing and lubricant adjustment could provide for a faster and cheaper alternative. In this work, a three-disk rotor was examined analytically for nonlinear rotordynamic behavior due to an unbalance-driven rub. The rotordynamic solution was obtained using nonlinear and steady state finite element models to demonstrate the effect of the rub impact on the dynamic response of the machine. A thermoelastohydrodynamic (TEHD) model of tilting pad journal bearing performance was also used to study the possible removal of the rub impact by making minor adjustments to bearing parameters including preload, clearance, pad orientation, and lubricant properties. Gas-expanded lubricants (GELs), tunable mixtures of synthetic oil and carbon dioxide that have been proposed as a means to provide control in bearing-rotor systems, were also considered for their possible role in controlling the rub. The TEHD model provided a range of bearing inputs to the rotor models in the form of stiffness and damping coefficients. Results from the rotordynamic analyses included an assessment of critical speeds, peak rotor displacements, and vibration characteristics. Individual bearing parameter adjustments were found to have smaller, though still significant effects on the response of the machine. Overall, it was found that by adjusting a combination of these bearing parameters, the peak displacement of the rotor could be reduced by large enough amounts to remove the rub impact in the machine, hence providing a simple approach to solving rub impact problems in rotating machinery caused by excessive vibration.

Rotor Dynamics Behavior of Tilting Pad Bearing Supported Turbo-Expander Considering Temperature Gradient

This paper dedicates on the rotor dynamics behavior analysis on a tilting pad bearing supported turbo-expander rotor system considering temperature gradient. Both numerical and experimental investigations are conducted intensively. The influence of the temperature gradient is modeled as the change of the lubrication oil viscosity and the length variation of the clearance due to the cryogenic thermal expansion of the journal. The analytical expressions of the tilting pad bearing oil-film force are then amended and substitute into the lumped parameter model of the turbo-expander rotor dynamics. Linear analysis based on this model indicates that the existence of the temperature gradient can stabilize the turbo-expander rotor system to an extent, while the nonlinear analyses reveal that the temperature gradient will advance the occurrence of the quasi-periodic motion and break the equilibrium of the vibration between the expander side and the compressor side. Furthermore, an experimental system is established and the experimental results show that the temperature of the tilting pad bearing is influenced by the environment temperature greatly; the spectrum of the displacement of the rotor is dominated by the synchronous frequency of the impellers and bearings. The experiment results also observe the vibration amplitude decreases when the environment temperature gets down and grows when the rotating speed increases. At the same time, the sensitivity of the vibration amplitude versus rotating speed decreases as the environment temperature rises, and vice versa.

Modeling a two-span rotor system based on the Hamilton principle and rotor dynamic behavior analysis

A nonlinear dynamic model of a two-span rotor system is constructed based on the Hamilton principle and the finite element method. The Musznyska model and the short bearing model are employed to describe the nonlinear seal force and oil-film force. The fourth-order Runge-Kutta method is used to calculate the numerical solutions. The bifurcation diagrams, time-history diagrams, phase trajectories, and Poincare maps are presented to analyze the dynamic behavior of the bearing center and the disk center in the horizontal direction. The numerical results indicate that the rotational speed, the nonlinear seal force, the oil-film force, and the stiffness of the coupling have a significant effect on the stability of the rotor system. The dynamic behavior of the two-span rotor system is more complicated when impacted by the nonlinear seal force and oil-film force.

Rotordynamic Behavior of 225 kW (300 HP) Class PMS Motor–Generator System Supported by Gas Foil Bearings

This paper presents the dynamic behavior of a 225 kW class (300 HP), 60,000 rpm, permanent magnet synchronous (PMS) motor–generator system supported on gas foil bearings (GFBs). The rotor of a 225 kW PMS motor is supported by two identical gas foil journal bearings (GFJBs) and one pair of gas foil thrust bearings (GFTBs). The total weight and axial length of the coupled rotors are 272 N and 1042 mm, respectively. During the speed-up test to 60,000 rpm, unexpected large subsynchronous rotor motions appear at around 120–130 Hz above 35,040 rpm. After disassembling the motor, an inspection of the top foils of the GFJBs reveals significant rotor rubbing. Thus, the GFJBs are redesigned to have a smaller load capacity by reducing their axial length to 45 mm. In addition, three 50 μm thick shims are installed in the GFJBs at 120 deg intervals for reducing the swirl speed of air and producing bearing preloads. The modification delays the onset speed of subsynchronous motions to 43,200 rpm and decreases the amplitude of the subsynchronous motions from 20 to 15 μm. These results indicate that the modification improves the stability margin of the high-speed rotor system with increasing stiffness and damping. In addition, the logarithmic decrement trends are in good agreement with the test results.

Turbocharger Synchronous Vibration Control on High Speed Balancer: Test and Prediction

Current trends for advanced automotive engines focusing on downsizing, better fuel efficiency, and lower emissions have led to several changes in turbocharger bearing systems design, and technology. Automotive turbochargers are running faster under high engine vibration level. Vibration control is becoming a real critical issue and turbocharger manufacturers are focusing more and more on new and improved balancing technology. This paper deals with turbocharger synchronous vibration control on high speed balancers. In a first step the synchronous rotordynamics behavior is identified. The developed fluid bearing code predicts bearing rotational speed (in case of fully floating design), operating inner and outer bearing film clearances and bearing force coefficients. A rotordynamics code uses this input to predict the synchronous lateral dynamic response of the rotor-bearing system by converging with bearing eccentricity ratio. The rotor-bearing system model is validated by shaft motion test data on high speed balancer (HSB). It shows that only one of the peaks seen on the synchronous G level plot collected in a high speed balancer can be explained by rotordynamics physics. A step-by-step structural dynamics model and analysis validated by experimental frequency response functions provides robust explanations for the other G level peaks. The synchronous vibration response of the system “turbocharger-HSB fixture” is predicted by integrating the predicted rotordynamics rotational bearing loads on the structural dynamics model. Numerous test data show very good correlation with the prediction, which validates the developed analytical model. The “rotordynamics—structural dynamics model” allows deep understanding of turbocharger synchronous vibration control, as well as optimization of the high speed balancer tooling.

The Impact of Hole Depth on the Rotordynamic and Leakage Characteristics of Hole-Pattern-Stator Gas Annular Seals

Test results are presented for rotordynamic characteristics of hole-pattern-stator annular gas seals at a 70 bar supply pressures with a running speed of 20,200 rpm. Leakage results are presented for these conditions and the additional speeds: 10,200 and 15,300 rpm. Hole-depth is the principal test variable of interest. Most published annular test data for these seals have a hole diameter of 3.175 mm and a hole depth of 3.302 mm. For this work, with the 3.175 mm hole diameter, additional results are presented for shallow (1.905 mm) and deep (6.604 mm) hole depths. Test results show a pronounced dependence of leakage and rotordynamic behavior on hole depth. Test results show much better leakage performance for the shallow-hole-depth seal in both leakage and rotordynamic performance. Compressor manufacturers and users will need to decide whether this observed performance improvement is worth trying in real machines.

A Systematic Approach to Estimate the Impact of the Aerodynamic Force Induced by Rotating Stall in a Vaneless Diffuser on the Rotordynamic Behavior of Centrifugal Compressors

One of the main challenges of the present industrial research on centrifugal compressors is the need for extending the left margin of the operating range of the machines. As a result, interest is being paid to accurately evaluating the amplitude of the pressure fluctuations caused by rotating stall, which usually occurs prior to surge. The related aerodynamic force acting on the rotor can produce subsynchronous vibrations, which can prevent the machine's further operation, in case their amplitude is too high. These vibrations are often contained due to the stiffness of the oil journals. Centrifugal compressor design is, however, going towards alternative journal solutions having lower stiffness levels (e.g., active magnetic bearings or squeeze film dampers), which will be more sensitive to this kind of excitation: consequently, a more accurate estimation of the expected forces in the presence of dynamic external forces such as those connected to an aerodynamically unstable condition is needed to predict the vibration level and the compressor operability in similar conditions. Within this scenario, experimental tests were carried out on industrial impellers operating at high peripheral Mach numbers. The dedicated test rig was equipped with several dynamic pressure probes that were inserted in the gas flow path; moreover, the rotor vibrations were constantly monitored with typical vibration probes located near the journal bearings. The pressure field induced by the rotating stall in the vaneless diffuser was reconstructed by means of an ensemble average approach, thus defining the amplitude and frequency of the external force acting on the impeller. The calculated force value was then included in the rotordynamic model of the test rig: the predicted vibrations on the bearings were compared with the measurements, showing satisfactory agreement. Moreover, the procedure was applied to two real multistage compressors, showing notable prediction capabilities in the description of rotating stall effects on the machine rotordynamics. Finally, the prospects of the proposed approach are discussed by investigating the response of a real machine in high-pressure functioning when different choices of journal bearings are made.

Rotordynamics of a Two-Phase Flow Twin Screw Pump

Twin screw pumps are positive displacement machines. Two meshing screws connected by timing gears push the fluid trapped in the screw cavities axially from suction to discharge. Available steady state hydraulic models predict pump performance and axial pressure distribution in the chambers in single- and two-phase flow conditions. However, no model is available for their rotordynamics behavior. Due to the helix angle of the screw, the pressure distribution around the rotor is not balanced, giving rise to both static and dynamic lateral forces. The work presented here introduces a starting point for rotordynamic analysis of twin screw pumps. First, we show that the screw rotor's geometry can be represented by axisymmetric beam elements. Second, we extend the steady state hydraulic model to predict both the static and dynamic lateral forces resulting from the unbalanced pressure field. Finally, hydraulic forces are applied to the rotor to predict static, synchronous, and nonsynchronous responses. Predictions of the dynamic pressure were compared to measurements from the literature and were found to be in good agreement.