Whirl Frequency Research Articles

Dynamic analysis for the thresher of a combined harvester subjected to stalk winding

In this paper, the whirl characteristics and instability of the thresher in a combined harvester subjected to stalk winding are investigated. Using the spinning Timoshenko beam assumptions, the governing equations of motion for the thresher-stalk system are obtained via Hamilton’s principle. Considering the elastic end supports, the exact and approximated dimensionless whirl frequency equations are derived. On the basis of the obtained formulas, both whirl characteristics and stability of the thresher-stalk system are investigated numerically, and the validation of the present analysis is conducted by comparing with the results available in the literature. In addition, the effects of mode number, mass ratio, eccentric distance, shear deformation, and slenderness ratio on the critical spinning speed, stability, and instability boundary are deeply discussed. The results show that the whirl characteristics and stability of the thresher-stalk system are strongly dependent on the stalks wound around the thresher.

A Nonlinear Rotordynamics Model for Automotive Turbochargers Coupled to a Physical Model for a (SEMI) Floating Ring Bearing System

Abstract Automotive turbochargers (TCs) use an engine oil lubricated bearing system to produce acceptable performance (as per the engine volumetric efficiency) and proven reliability. However, the bearings also cause TC rotordynamic responses that are rich in subsynchronous whirl motions through reaching stable limit cycles. The paper describes the lubrication model for a finite length semi-floating ring bearing (SFRB) system and its coupling to the rotor and ring structure models for prediction of both linear and nonlinear system responses and their characterization in terms of motion amplitudes and whirl frequency content. The SFRB model includes a thermal energy transport network for the inner and outer films in both radial bearings and the thrust bearings located on the end sides of the ring. The large temperature difference between the hot shaft and a cold housing induces a three-dimensional thermal gradient in the fluid films and the floating ring, further exacerbated by the heat generated from drag power losses in the inner films adjacent to the rotor. The temperature gradients affect the lubricant viscosity and the bearing system operating clearances. The integration of the rotor and bearing system (RBS) equations of motion accounts for the SFRB nonlinear forces and starts from static equilibrium position, if existing. The model predictions of nonlinear behavior are accurate when benchmarked to a set of measurements procured in a gas stand test rig. The analysis also investigates the influence of the bearing inner clearance and rotor mass imbalance distribution on the onset, persistence and severity of SSV.

Rotordynamics of an accelerating rotor supported on aerodynamic journal bearings

Aerodynamic bearings are being employed in several high-speed micro-turbomachines, where frequent acceleration/deceleration of the rotor is common. Keeping the whirl amplitude of the rotor within acceptable limits is a must for the safety of the rotor-bearing system. This paper presents, for the first time, the dynamics of an accelerating rotor supported on Aerodynamic Journal Bearings. Due to unbalance, the rotordynamics between the spin and translational coordinates get nonlinearly coupled. The paper utilizes these equations to plan suitable acceleration-schedules to limit the whirl amplitude and frequency as low as possible. It is found that the acceleration should be scheduled in at least three regimes: the first with the maximum torque, the second with the minimum torque, and the last with an intermediate torque.

Investigations of the static and dynamic performances of an annular seal with a tilted rotor by transient CFD methods with dynamic mesh

A transient computational fluid dynamics (CFD) method based on dynamic mesh is adopted to investigate the static and dynamic performances of an annular seal with a tilted rotor. The reliability of the CFD method is verified by the experimental data, and then the variations of dynamic performance and sealing performance under different length-diameter ratios and misalignment angles are investigated by the method. The results show that the leakage flowrate of the tilted annular seal decreases with the increasing length-diameter ratio, while the whirl frequency ratio increases with the length-diameter ratio. The increase of misalignment angle has little detrimental effect on the stability of long shaft system in the three misalignment modes. In addition, the misalignment mode for the tilted position at the half-length of the annular seal can reduce the leakage flowrate.

A Study on Dynamic Behaviour of Thermally Distributed Exponentially Graded Rotor System with Induced Porosities

The present work deals with the dynamic analysis of exponential law-based functionally graded (FG) rotor-bearing systems. The effect of thermal gradation and porosity on dynamic characteristics of FG rotor shafts has been studied first time, using exponential law with a novel two-nodded FG rotor element based on Timoshenko beam theory (TBT). Porous material properties are assorted using exponential law and thermal gradation across the cross section of the FG shaft using exponential temperature distribution (ETD). The effects of temperature and porosity on natural frequencies and whirl frequencies are studied. It has been observed that there is a significant reduction in natural frequencies and whirl frequencies with an increase in volume fraction of porosity and temperature. Attempts have been made to obtain suitable reasons for the behaviours based on the material properties. Furthermore, the steady-state and transient vibration responses have been simulated using the Houbolt time marching technique for the ceramic-based FG rotor shaft system. The result shows the maximum amplitude of the steady-state and transient vibration responses is increased, and the critical speed of the FG rotor system shifts towards the left with the increase in volume fraction of porosity and temperature.

Whirl dynamics of an axially functionally graded liquid-filled rotor considering shear deformation and rotary inertia

In this study, whirl characteristics and stability of an axially functionally graded (AFG) liquid-filled rotor are investigated. The rotor is modeled based on the spinning Timoshenko beam theory. The governing equations for flexural vibration are derived via Hamilton’s principle. For pinned–pinned AFG liquid-filled rotor, the analytical solutions are derived for both the exact whirl frequency equation and the stability model. To validate the present formulations, comparative studies by numerical solutions available in the literature are conducted. Some numerical examples are performed to investigate the effects of gradient parameter, mass ratio, cavity ratio, rotary inertia, and shear deformation on the whirl speed, the critical spinning speed, and the stability of the AFG liquid-filled rotor system. The results show that these parameters have noticeable influences on dynamic behavior and stability of the rotor system. In particular, the rotary inertia and shear deformation play an important role in the stability analysis for different length rotors.

Gyroscopic effect on a scaled rotor-bearing system

This paper deals with scale-fullscale models of rotor-bearing systems on rotating condition. The investigation is focused on the gyroscopic effect, which causes forward and backward whirl frequencies. When the rotor-bearing system is scaled proportionally in its dimension (height, length, and width), the scaling factor of whirl frequencies can be derived. It depends on the ratio between a transverse and polar moment of inertia on its rotating axis called the gyroscopic factor. The experimental study is conducted to validate it on three scaled rotor-bearing systems, which shifts its disc from the middle of the shaft on the scale of 1:1, 2:1, and 3:1 to clearly show the gyroscopic effect on first bending natural frequency. The scaling factor is then validated using the Campbell diagram by finding its critical speed. From this critical speed, the whirl frequencies along the range of the full-scale model speed can be obtained. The result also shows that the scaling factor remains the same whether it is at rotation or rest condition. Consideration must be made on the effect of the structural design, that is, blade and support, because of its unsymmetric stiffness that can cause backward whirl frequencies. The bearing stiffness must be ensured to be scaled proportionally, especially on journal bearing cases. This finding can be used by engineers to deal with scaling method implementation on rotating machinery design.

Flow analysis of the hole diaphragm labyrinth seal by the CFD method

The computational fluid dynamics (CFD) method is an efficient way to reduce the obstacles in fluid studies. Good use of the CFD method on the seal-rotor system area was proven again in this paper. The flow analyses of a new type of annular seal, the hole diaphragm labyrinth seal (HDLS), were studied by Fluent. Complex working conditions were considered to further understand the flow field variation trend. The results show that the maximum velocity component was obtained in the middle cavities perpendicular to the whirl direction. The velocity distributions before the first cavity and behind the last cavity show a different trend under a 50 Hz whirl frequency than that of other whirl frequencies, which can be described by the most apparent throttling effect of the first blade and the diffuse effect behind the last blade. It is helpful to further understand different annular seal characteristics by detailed fluid information analyses.

Rotordynamic analysis and leakage performance study of a hole diaphragm labyrinth seal using the CFD method

Improving rotor stability and reducing leakage at various working parts of gas turbines is an important topic that experts and scholars have always pursued. In this article, a new type of hole-diaphragm-labyrinth seal (HDLS) structure is obtained by introducing an additional damping source. The HDLS structure is built by the traditional diaphragm labyrinth seal (DLS) with a single hole added to the diaphragm. The simulation under different working conditions of three different seal structures is carried out by ANSYS Fluent. Subsequently, the tangential force under different whirl frequencies was obtained. Meanwhile, the empirical parameters in Childs’ model of the dynamic performance of rotor-seal structures were obtained. The results show that the HDLS achieved a 71.4% improvement in the direct stiffness coefficient and a 71.8% improvement in the direct damping coefficient of the traditional labyrinth seal. The leakages of HDLS at different rotation speeds were lower than those of DLS by 1.53% and LS by 3.64%. The HDLS can be comprehensively considered the most promising seals among LS, DLS, and HDLS not only by stability but also by mass flow leakage in the rotor system.

Dynamic Stiffness Matrix Approach to Free Vibration Analysis of Functionally Graded Rotor Bearing System Subjected to Thermal Gradients.

The dynamic stiffness matrix (DSM) method, an analytical method that provides exact solutions, has been used for the first time for the free vibration analysis of a functionally graded (FG) rotor bearing system subjected to temperature gradients and to investigate its application to FG rotors. The material gradation occurs based on the power law between the inner metal core and the outer ceramic rich layer of the FG rotor. The temperature gradation follows the Fourier law of heat conduction which leads to non-linear temperature distribution (NLTD) in the radial direction of the FG rotor. The development of the DSM formulations for Timoshenko FG rotor elements using the governing equations derived from translational and rotational equilibrium conditions is the novelty of the present work. The DSM of the FG rotor elements, rigid disk and linear isotropic bearings are assembled to obtain the global dynamic stiffness matrix of the FG rotor bearing system. The natural whirl frequencies are computed from the global DSM using the Wittrick–William algorithm as a root searching technique. The natural and whirl frequencies are validated with the results available in the literature and the exactness of the DSM method has been exemplified.

All-round responses and boundaries of a shaft and dry friction damper assembly

A helicopter tailrotor driveline is equipped with dampers to suppress the resonance of transmission shafts. A dry friction damper and a slender shaft form an assembly The comprehensive understanding of different responses of the assembly provides the basis for a more reasonable parameter configuration of the damper. To this end, the assembly is modeled as a dynamic system with rub impact between a rotor with continuous parameter distribution and a suspended elastic stator. Boundaries of rub impact and condition of Hopf bifurcation are derived. Pinned natural frequencies and backward whirl frequencies are solved by analytical deduction. The critical speed for triggering a backward whirl with consideration of eccentric excitation is obtained by the multiple scale perturbation method. Then, all-round responses and boundaries are solved by both analytical and numerical approaches based on the parameters of a helicopter. The effects of friction coefficient, internal damping, suspension stiffness, variable impact stiffness, etc. on the boundaries are presented in detail with the discussion of relationships between physical reasons and analytical conditions. Results show that smaller impact stiffness difference and friction coefficient, a greater mass damping ring, moderate suspension stiffness, greater internal and external damping are the key factors for the damper design. This study has reference significance not only for the assembly in a helicopter but also for similar mechanical systems.

Simulation and Flow Analysis of the Hole Diaphragm Labyrinth Seal at Several Whirl Frequencies

The seal is designed to reduce leakage and improve the efficiency of gas turbine machines, and is an important technology that needs to be studied in gas turbine design. A series of seals were proposed to try to achieve this goal. However, due to the complex fluid dynamic performance of the seal-rotor system, the seal structure can obtain both the best leakage performance and best rotordynamic performance. This paper presents a detailed flow analysis of the hole diaphragm labyrinth seal (HDLS) at several whirl frequencies and several rotation speeds. The pressure drop, velocity, turbulence kinetic energy and leakage performance of the HDLS were discussed by simulations. An interesting exponential–type relationship between rotation speeds and leakage flow at different whirl frequencies was observed by curve fitting technology. A reverse flow rate was proposed to describe such an unusual phenomenon. Such a relationship can be used to further establish the leakage model of the HDLS and other similar seals.

Experimental Research on Bit Side Force and Anti-deviation in Vertical Well

Based on dimensional method, the similarity criterion of simulation test was derived. The experimental apparatus was established in accordance with the similarity criterion. A series of experiments with different drilling parameter were carried out. The experimental results show that with rotary speed rise, the bit side force fluctuation increased, however, the vibration energy at the dominant frequency decreased, at low rotary speed, the dominant frequency of the bit side force is equal to rotation frequency of drill string, and at high rotary speed, the dominant frequency is consistent with the backward whirl frequency of drill string, which is about 3 times of rotation frequency, moreover, as WOB rise, the vibration energy at dominant frequency increased, while the total wave energy decreased, and the critical rotary speed value at the jump process of the dominant frequency increased. Furthermore, for the purpose of deviation controlled straight drilling, there is the problem of optimization grouping between WOB and rotary speed.

A Nonhomogeneous Bulk Flow Model for Gas in Liquid Flow Annular Seals: An Effort to Produce Engineering Results

Abstract Seals in multiple-phase rotordynamic pumps must operate without compromising system efficiency and stability. Both field operation and laboratory experiments show that seals supplied with a gas in liquid mixture (bubbly flow) can produce rotordynamic instability and excessive rotor vibrations. This paper advances a nonhomogeneous bulk flow model (NHBFM) for the prediction of the leakage and dynamic force coefficients of uniform clearance annular seals lubricated with gas in liquid mixtures. Compared to a homogeneous bulk flow model (HBFM), the current model includes diffusion coefficients in the momentum transport equations and a field equation for the transport of the gas volume fraction (GVF). Published experimental leakage and dynamic force coefficients for two seals supplied with an air in oil mixture whose GVF varies from 0% (pure liquid) to 20% serve to validate the novel model as well as to benchmark it against predictions from a HBFM. The first seal withstands a large pressure drop (∼38 bar) and the shaft speed equals 7.5 krpm. The second seal restricts a small pressure drop (1.6 bar) as the shaft turns at 3.5 krpm. The first seal is typical as a balance piston whereas the second seal is found as a neck-ring seal in an impeller. For the high pressure seal and inlet GVF = 0.1, the flow is mostly homogeneous as the maximum diffusion velocity at the seal exit plane is just ∼0.1% of the liquid flow velocity. Thus, both the NHBFM and HBFM predict similar flow fields, leakage (mass flow rate), and drag torque. The difference between the predicted leakage and measurement is less than 5%. The NHBFM direct stiffness (K) agrees with the experimental results and reduces faster with inlet GVF than the HBFM K. Both direct damping (C) and cross-coupled stiffness (k) increase with inlet GVF < 0.1. Compared to the test data, the two models generally under predict C and k by ∼25%. Both models deliver a whirl frequency ratio (fw) ∼ 0.3 for the pure liquid seal, hence closely matching the test data. fw raises to ∼0.35 as the GVF approaches 0.1. For the low-pressure seal, the flow is laminar; the experimental results and both NHBFM and HBFM predict a null direct stiffness (K). At an inlet GVF = 0.2, the NHBFM predicted added mass (M) is ∼30% below the experimental result while the HBFM predicts a null M. C and k predicted by both models are within the uncertainty of the experimental results. For operation with either a pure liquid or a mixture (GVF = 0.2), both models deliver fw = 0.5 and equal to the experimental finding. The comparisons of predictions against experimental data demonstrate that the NHBFM offers a marked improvement, in particular for the direct stiffness (K). The predictions reveal that the fluid flow maintains the homogeneous character known at the inlet condition.

Fundamental frequency analysis of a thermally loaded corroded exponentially graded rotor bearing system

ABSTRACT Corrosion is an unintentional degradation of a material, which transpires due to harsh environmental conditions. It is a surface phenomenon that results in loss of material, which affects the dynamic characteristics and structural integrity of any structure. Fundamental frequency analysis of a corroded functionally graded (FG) rotor-bearing system using finite element method (FEM) for flexural vibrations is presented in work. A functionally graded (FG) shaft, Stainless Steel as the inner-metal core and Zirconia as the outer-ceramic layer is considered. Exponential gradation law is followed to assign the material properties along the radial direction of the shaft. The exponential temperature distribution (ETD) method based on Fourier law of heat conduction has been used for the temperature distribution across the cross-section of the FG shaft. Finite element formulations of a corroded FG shaft element have been developed using Timoshenko beam theory. An FE code is developed to compute the natural and whirl frequencies of an FG rotor-bearing system for various parameters such as corrosion length, position, depth and thermal gradients to investigate the influence of corrosion on fundamental frequencies. It has been investigated that there is a significant influence of corrosion parameters on the natural and whirl frequencies of an FG rotor-bearing system.

Porous Gas Journal Bearings: An Exact Solution Revisited and Force Coefficients for Stable Rotordynamic Performance

Having come of age, gas film bearings enable high-speed oil-free (micro) rotating machinery with gains in efficiency and reliability, longer maintenance intervals, and a reduction in contaminants released to the atmosphere. Among gas bearing types, porous surface gas bearings (PGBs) have proven successful for 50+ years and presently are off-the-shelf mechanical elements. This paper reviews the literature on PGBs since the 1970s and reproduces an exact solution for the performance of cylindrical PGBs. Both the analytical model and an accompanying finite-element (FE) model predict the performance for two PGBs, a commercially available 76 mm diameter bearing and a smaller 25 mm diameter laboratory unit whose experimental performance is available. As expected, the FE model results reproduce the analytical predictions obtained in a minuscule computing time. For a set external supply pressure, as the radial clearance increases, the flow rate through the bearing grows until reaching a peak magnitude. The PGB load capacity is a fraction of the product of the set pressure difference (pS − pa) and the bearing projected area with a significantly large centering static stiffness evolving over a narrow region of clearances. Operation with shaft speed enhances the bearing load capacity; however, at sufficiently high speeds, significant magnitude cross-coupled forces limit the stable operation of a PGB. At constant operating shaft speed, as the whirl frequency grows, the bearing effective stiffness (Keff) increases, while the effective damping (Ceff) becomes positive for whirl frequencies greater than 50% shaft speed. Similar to a plain hydrodynamic journal bearing, the PGB is prone to a half-frequency whirl, albeit the system natural frequency can be high, mainly depending on the external supply pressure. In essence, for the cases considered, PGBs are linear mechanical elements whose load capacity is proportional to the journal eccentricity.

Control of self-excited rotor disturbances in three-lobe journal bearing space using a couple stress lubricant

In this study, the dynamic behavior of three-lobe noncircular hydrodynamic journal bearings lubricated with couple stress fluid is presented. Currently, application of journal bearings with different geometries such as lobed bearings and using non-Newtonian lubricants, including various additives, are the common suggested solutions by tribology researchers to improve the performance of rotor-journal bearing systems. Due to the incompatibility of new lubricant behaviors including polymer chains or other types of suspended particles with considerable size from the classic Navier–Stokes equations, different models such as couple stress fluid theory have been proposed to evaluate their performance. To investigate the self-excited rotor disturbances caused by various factors, like operating in critical speeds, simultaneous solution of governing Reynolds equation of oil lubricant and the rotor motion equations has been done. The result of dynamic analyses using linear and nonlinear approaches indicates the possibility of enhancing the stiffness and damping ability of the three-lobe bearings by strengthening the couple stress properties of lubricant. It is seen from the results that by increasing the couple stress parameter of oil, the type of rotor perturbations in the bearing clearance space changes from divergent oscillations, as a leading cause of collision between the rotor surface and the bearing shell, to limit cycle motion around the rotor stable position or convergent fluctuations to the static equilibrium point. This improvement in the stability range of three-lobe journal bearings using a couple stress lubricant appears in the results of a linear analysis model in terms of increasing the critical mass parameter and decreasing the whirl frequency ratio. In addition, bifurcation diagram results show that using noncircular three-lobe bearings instead of circular types and replacing the Newtonian lubricant with the couple stress fluid improve the dynamic stability of rotor- bearing supports.

The Characteristics of Rotordynamic Forces generated by Mechanical Seals

Evaluating rotordynamic reaction force caused by whirl excitation is very effective technique in order to investigate the dynamic properties of the mechanical seal. Especially, it is easy to understand the stability of the rotating component by investigating rotordynamic tangential force. In this paper, the stability of the mechanical seal was discussed by measuring the rotordynamic tangential force depending on the whirl frequency. Judging from the result, the mechanical seal used for this study has stable performance that the tangential force always acts against whirling direction at 1,000 rpm of rotational speed and up to 30 Hz of whirl frequency. Furthermore, whirl frequency ratio and effective damping were calculated using cross-coupled stiffness and direct damping which were obtained by solving quaternary linear equations after whirl excitation test. Whirl frequency ratio was always bellow 1.0, and effective damping was also always positive at all frequencies. Those results indicate that the mechanical seal definitely generate the force to stabilize this whirl vibration.

Numerical calculation of the rotordynamic coefficients of a LOX-Turbopump Seal for the LUMEN LOX/LNG demonstrator rocket engine

In rocket engines seals are one of the key elements for a successful turbopump operation. They are not only responsible for reducing the leakage but they also have a huge influence on the rotordynamic behaviour. Depending on the design, the seal on a turbopump impeller can damp or excite rotordynamic instabilities. In this work we present transient 3D CFD simulations of straight gap seals, which are used to determine the rotordynamic coefficients. After that the numerical simulation is extended to characterize the influence of the impeller to housing clearance and swirl brakes on the rotor dynamic coefficients. We show that the inertia term cannot be neglected for the investigated nominal rotational speed of the rotor and high absolute whirl frequency ratios especially for the rotor housing clearance. Finally the coefficients calculated by numerical simulation are used in the equations of motion for a floating ring seal and the equations are solved numerically to do a sensitivity study of a floating ring to variations in axial force, rotor displacement and floating ring mass.

Dynamics and stability analysis of an axially functionally graded hollow rotor partially filled with liquid

The dynamics and stability of an axially functionally graded (AFG) hollow rotor partially filled with liquid are performed in this paper. Based on the Euler-Bernoulli beam theory, the governing equation of transverse vibration of an AFG liquid-filled rotor is derived using Hamilton principle. For the exponentially graded liquid-filled rotor with simply supports, the exact whirl frequency equation and the stability prediction model are obtained analytically. The validity of the present analysis is demonstrated by comparing with the existing data in the literature. Using the obtained analytical models, the effects of parameters including gradient parameter, mass ratio and cavity ratio on the whirl speed, critical spinning speed and system stability are put into evidence through a set of parametric studies. The results show that these parameters play an important role in the whirl speed and stability of the rotor system. Furthermore, the obtained results also indicate that the critical spinning speed of an AFG liquid-filled rotor is strongly dependent on the mass ratio and gradient parameter.