Squeeze Film Damper Model Research Articles

Squeeze Film Damper Suppression of Thermal Bow-Morton Effect Instability

Abstract The Morton effect (ME) is a synchronous vibration problem in turbomachinery caused by the nonuniform viscous heating around the journal circumference, and its resultant thermal bow (TB) and ensuing synchronous vibration. This paper treats the unconventional application of the SFD for the mitigation of ME-induced vibration. Installing a properly designed squeeze film damper (SFD) may change the rotor's critical speed location, damping, and deflection shape, and thereby suppress the vibration caused by the ME. The effectiveness of the SFD on suppressing the ME is tested via linear and nonlinear simulation studies employing a three-dimensional (3D) thermohydrodynamic (THD) tilting pad journal bearing (TJPB), and a flexible, Euler beam rotor model. The example rotor model is for a compressor that experimentally exhibited an unacceptable vibration level along with significant journal differential heating near 8000 rpm. The SFD model includes fluid inertia and is installed on the nondrive end bearing location where the asymmetric viscous heating of the journal is highest. The influence of SFD cage stiffness is evaluated.

Passive vibration reduction of a squeeze film damper for a rotor system with fit looseness between outer ring and housing

Clearances between bearing outer ring and sleeve can generally be maintained to provide a margin for the thermal expansion of the bearings. However, temperature variation, improper assembly and long-term vibration can enlarge the clearances and accelerate mechanical wear, leading to what is known as the fit looseness fault. Therefore, it is important to study a fit looseness fault model and investigate how to control the vibration coming from the fit looseness fault. In this paper, a Jeffcott rotor system with three disks was modeled as a single unit. A fit looseness model was applied in the whole rotor model to study the contact problems and response characteristics using a numerical integration method. Then, a squeeze film damper model was applied to assess the vibration reduction effects on the whole rotor system with the fit looseness fault. By comparing the results of the fit looseness fault without squeeze film damper and with squeeze film damper, it is found that the squeeze film damper can reduce nonlinear vibration responses effectively generated by the fit looseness fault for the nonlinear contact. This research work contributes to understanding the mechanism of fit looseness fault and controlling strong nonlinear vibration responses due to the fit clearances.

Analytical investigation of air squeeze film damping for bi‐axial micro‐scanner using eigenfunction expansion method

Squeeze film damping is an important factor in the dynamic stability of the bi‐axial microelectromechanic systems. The purpose of this article is to provide an analytical solution for calculating the air squeeze film damping in bi‐axial torsional micro‐mirrors, considering the changes of pressure distribution in rotation around different axes. This is the first time that has been done for a circular micro‐mirror. One of the advantages of bi‐axis micro‐scanners compared to single‐axis types is the larger space covered by the scan in different axes. To calculate the pressure distribution in different rotation of micro‐mirror and also to achieve the squeeze film damping torque, Reynolds nonlinear equations have been used in polar coordinates, which have been solved by Taylor series expansion and also by eigenfunction expansion method. The results are verified with previous research in single‐axis mode. The results also show that with increasing gap between the micro‐mirror and the scanner substrate, the coefficient squeeze film damping will be less, but if the air gap is not changed, increasing the micro‐mirror radius will increase the damping coefficient. When the radius of micro‐mirror decrease, in order for the damping coefficient to be higher, the angle of rotation of the micro‐mirror must be greater too. The results of this article can be used for accurate modeling of squeeze film damping in micro‐scanners, control of dual‐axis torsion micro‐mirrors, and improvement of their efficiency.

Auxiliary bearing squeeze film dampers for magnetic bearing supported rotors

Auxiliary bearings (AB) support the rotor and protect the magnetic bearing (AMB) system when the AMB is disabled due to power loss or excessive loads. The paper demonstrates that installing a damping device along with the AB can yield extended AB fatigue life, protect the AMB, and reduce vibration, contact force and AB heating. A squeeze film damper (SFD) is an energy dissipation device that has been widely used in the turbo-machinery industry, and as demonstrated in the paper can also work effectively in combination with an AB. Usually, the SFD implements a supply groove to ensure adequate lubricant flow into the film lands. The supply groove can provide significant added mass coefficients and significantly influence the overall impedance of the SFD. Past literature has analyzed the transient response of the rotor dropping onto AB's with squeeze film dampers, none though have considered the influence of the SFD's center groove and its added mass effect on rotor's drop behavior. This paper develops a high fidelity finite element grooved SFD model considering the fluid inertia, and an effective groove clearance is used following the practice appearing in the literature. SFD force coefficients are benchmarked with results of a linear fluid inertia, bulk-flow model developed in the literature, before including them in the rotor – AB system model. The SFD model is integrated into a high fidelity nonlinear auxiliary bearing (angular contact ball bearing) model, which considers the movements, contact force, stress, and temperature of bearing balls, the inner race and outer race. The instantaneous reaction forces from the SFD are calculated with a finite element based solution of Reynold equation at each time step due to the intermittent and large sudden loads. The flexibility of the rotor is included utilizing a Timoshenko beam, finite element model. The fatigue life of the auxiliary bearing when integrated into the SFD is also calculated based on the rain flow counting method. The influence of the added mass of the SFD on the rotor's drop behavior is demonstrated showing that the added mass increases the contact force and peak temperature and reduces the fatigue life of the AB. Therefore, the added mass effect of the SFD should be considered to avoid over predicting the AB fatigue life. The influence of the SFD clearance on the rotor's drop behavior is also studied showing that an optimal clearance exists for increasing the AB fatigue life. Too small of a clearance will yield excessive damping making the effective stiffness too large, and causing high contact forces. Too large of a clearance lowers damping which may lead to a destructive backward whirl. This paper provides key guidelines for auxiliary bearing damper system design.

Ball bearing turbocharger vibration management: application on high speed balancer

Today's modern internal combustion engines are increasingly focused on downsizing, high fuel efficiency and low emissions, which requires appropriate design and technology of turbocharger bearing systems. Automotive turbochargers operate faster and with strong engine excitation; vibration management is becoming a challenge and manufacturers are increasingly focusing on the design of low vibration and high-performance balancing technology. This paper discusses the synchronous vibration management of the ball bearing cartridge turbocharger on high-speed balancer and it is a continuation of papers [1–3]. In a first step, the synchronous rotordynamics behavior is identified. A prediction code is developed to calculate the static and dynamic performance of “ball bearing cartridge-squeeze film damper”. The dynamic behavior of balls is modeled by a spring with stiffness calculated from Tedric Harris formulas and the damping is considered null. The squeeze film damper model is derived from the Osborne Reynolds equation for incompressible and synchronous fluid loading; the stiffness and damping coefficients are calculated assuming that the bearing is infinitely short, and the oil film pressure is modeled as a cavitated π film model. The stiffness and damping coefficients are integrated on a rotordynamics code and the bearing loads are calculated by converging with the bearing eccentricity ratio. In a second step, a finite element structural dynamics model is built for the system “turbocharger housing-high speed balancer fixture” and validated by experimental frequency response functions. In the last step, the rotating dynamic bearing loads on the squeeze film damper are coupled with transfer functions and the vibration on the housings is predicted. The vibration response under single and multi-plane unbalances correlates very well with test data from turbocharger unbalance masters. The prediction model allows a thorough understanding of ball bearing turbocharger vibration on a high speed balancer, thus optimizing the dynamic behavior of the “turbocharger-high speed balancer” structural system for better rotordynamics performance identification and selection of the appropriate balancing process at the development stage of the turbocharger.

Identification of force coefficients in inter-shaft squeeze film dampers utilizing complex vector description and computational fluid dynamics

Squeeze film dampers (SFDs) have been widely used in high-speed turbomachinery to provide external damping to the system to improve stability. An inter-shaft damper (ISD) is a SFD-like device inserted between two shafts of a double spool aircraft engine to provide damping to both internal and external rotors. Compared to SFDs, analysis of ISDs is not well established because of added complexity of the mechanism. A new approach to model ISDs is reported in this paper, which converts an ISD model to a dynamically equivalent SFD model. A systematic analysis method for SFDs subjected to whirling motions of a non-circular orbit is established first because the whirling orbit of the equivalent SFD is always non-circular with two major frequency components. The analysis method utilizes complex vector descriptions of the motion and force, and frequency-domain analysis with double-sided frequency spectrums. The proposed approach is validated by comparing computational fluid dynamics (CFD) analysis results and damping coefficients obtained from the original ISD model and its equivalent SFD model. One significant advantage of the new approach is that all design rules and insights developed for SFDs can be directly applied to ISDs. In addition, the analysis method established in this work enables accurate calculation of damping coefficients of SFDs subjected to non-circular whirling motions, essentially all SFDs in in real operations including SFDs in test rigs.

The effect of lubricant inertia on fluid cavitation for high-speed squeeze film dampers

This work studies the effect of lubricant inertia on the fluid cavitation for partially sealed high-speed squeeze film dampers (SFDs) executing small amplitude circular-centered orbits (CCOs). The lubricant cavitation is modeled by both the Elrod algorithm and the Gumbel’s cavitation boundary condition to provide the comparison between the most common lubricant cavitation models. Additionally, the fluid inertia is integrated by adapting a finite-length SFD model for partially sealed dampers. The integrated SFD model is incorporated into a numerical simulation model and the results are validated by comparison with experimental data. The results of the analysis demonstrate that the fluid inertia effects significantly extend the cavitation region and influence the cavitation onset and the film reformation.

Selecting a squeeze film damper model for solving problems of gas turbine engine rotor dynamics

Various standard mathematical squeeze film damper (SFD) models that can be applied for solving rotor dynamics problems of gas turbine engines (GTE) are presently available. The accuracy of simulation is ultimately affected by the choice of a certain SFD formulation that depends on many factors geometry parameters, oil supply method, oil inlet pressure, operating conditions etc. The present contribution suggests some recommendations for selecting a particular mathematical SFD model and compares the simulation results obtained with different SFD formulations. To this end a point rotor system is numerically analyzed with the aid of the DYNAMICS R4 (www.alfatran.com) software.

A Study of Lubricant Inertia Effects for Squeeze Film Dampers Incorporated into High-Speed Turbomachinery

This work proposes a numerical model that incorporates the effect of lubricant inertia on the hydrodynamic pressure distribution, fluid film reaction forces, and the fluid velocity component profiles for finite-length open-ended squeeze film dampers (SFDs). Firstly, the thin film flow equations for the SFD in presence of fluid inertia effects are introduced. Furthermore, a small first-order perturbation by means of the expressions for the fluid film velocity components and the lubricant pressure distribution that are expanded in power series of the squeeze film Reynolds number is applied to the flow equations. Subsequently the developed lubricant flow equations are solved to develop expressions for the velocity component profiles and the hydrodynamic pressure distribution in SFDs. The pressure expression is numerically solved by using Gauss–Seidel method with finite difference discretization. Moreover, the fluid film reaction forces are determined by numerically integrating the hydrodynamic pressure expressions over the journal surface. Additionally, the proposed pressure distribution expression and the numerical SFD forces are incorporated into a simulation model and the simulation results are compared with the existing models in the literature under different operating conditions, including eccentricity ratios and inertia effects (i.e., Reynolds numbers). The simulation results demonstrate the significant influence of both convective and temporal (i.e., unsteady) lubricant inertia terms on the SFD hydrodynamic pressure distribution and the fluid film reaction forces. Furthermore, the proposed SFD model is incorporated into a multi-mass flexible rotordynamic model to evaluate the effect of SFD fluid inertia on the mass unbalance induced steady-state vibrations of the rotor and the nodal transient orbits by implementing finite element method and transient modal integration with predictor–corrector solver. The results of the analysis demonstrate the significant effect of fluid inertia on the resonance frequencies of the rotor and the steady-state vibration amplitudes and the transient orbits at the resonance zone.

Analytical solution for squeeze film damping of MEMS perforated circular plates using Green’s function

Squeezed air film between two closely spaced vibrating microstructures is the important source of energy dissipation and has profound effects on the dynamics of microelectromechanical systems (MEMS). Perforations in the design are one of the methods to model these damping effects. The literature reveals that the analytical modeling of squeeze film damping of perforated circular microplates is less explored; however, these microplates are also an imperative part of the numerous MEMS devices. Here, we derive an analytical model of transverse and rocking motions of a perforated circular microplate. A modified Reynolds equation that incorporates compressibility and rarefaction effects is utilized in the analysis. Pressure distribution under the vibrating microplate is derived by using Green’s function and also derived by finite element method (FEM) to visualize the pressure distribution under perforated and non-perforated areas of the microplate. The analytical damping results are validated with previous renowned analytical models and also with the FEM results. The outcomes confirm the potential of the present analytical model to accurately predict the squeeze film damping parameters.

Response of a Squeeze Film Damper-Elastic Structure System to Multiple and Consecutive Impact Loads

Squeeze film dampers (SFDs) are common in aircraft gas turbine engines, customized to provide a desired level of damping while also ensuring structural isolation. This paper presents measurements obtained in a test rig composed of a massive cartridge, an elastic structure, and an open-ends SFD with length L = 25.4 mm, diameter D = 127 mm, and radial clearance c = 0.267 mm. ISO VG 2 oil at room temperature lubricates the thin film. The measurements quantify the system transient response to sudden loads for motions departing from various static eccentricity displacements, es/c = 0–0.6. The batch of tests include recording the system response to (a) one single impact, (b) two (and three) impacts with an elapsed time of 30 ms in between, and (c) two or more consecutive impacts, without any delay, each with a load magnitude at 50% of the preceding impact. The load actions intend to reproduce, for example, a hard landing on an uneven surface or plunging motions from sudden contacts in a machine tool. The test system transient responses due to one or more impacts, each 30 ms apart, show the peak amplitude of motion (ZMAX) is proportional to the magnitude of applied load (FMAX). The identified system damping ratio (ξ) is proportional to the peak dynamic displacement as a linear system would show. Predictions of transient response from a physical SFD model accounting for fluid inertia correlate best with the experimental results as they produce greatly reduced peak dynamic motions when compared to predictions from a purely viscous SFD model. For the responses due to consecutive impacts, one after the other with no delay, the system motion does not decay immediately but builds to produce larger motion amplitudes than in the earlier cases. Eventually, as expected, after several oscillations, the system comes to rest. For an identical damper having a smaller clearance cs = 0.213 mm (0.8c), its damping ratio (ξs) is ∼1.3 to ∼1.7 times greater than the damping ratio for the damper with a larger film clearance (ξ). Hence, the experimentally derived (ξs/ξ) scales with (c/cs)2. The finding demonstrates the importance of manufacturing precisely the components in a damper to produce an accurate clearance.

A numerical molecular dynamics approach for squeeze-film damping of perforated MEMS structures in the free molecular regime

Accurate determination of the squeeze-film damping in rare air is crucial for the design of high-Q MEMS devices. In the past, for the MEMS structures with no perforations, there have been two approaches to treating the squeeze-film damping in rare air: the approach based on the continuum assumption and the approach using molecular dynamics (MD) method. The amount of squeeze-film damping can be controlled by providing perforations in microstructures. To model perforation effects on squeeze-film damping, many methods have been proposed. However, almost all the previous methods are based on the continuum assumption. Only one paper focuses on analytical modeling of squeeze-film damping of a perforated microplate using the MD method. Hutcherson and Ye (J Micromech Microeng 14:1726–1733, 2004) developed a novel MD method to model the squeeze-film damping in free molecular regime. The method possesses high computational efficiency. However, their work is valid only for non-perforated rectangular microplate. This paper presents a numerical MD approach for calculating the squeeze-film damping of a perforated rectangular plate and a perforated circular plate in free molecular regime. In Hutcherson and Ye’s work, the microplate is non-perforated. After each collision with the non-perforated plate, all the molecules are reflected to the substrate. In this paper, the plate is perforated. For the molecules in the air gap striking the surface of the perforated microplate, some of the molecules are reflected to the substrate. The rest leave the air gap through the perforations. This paper is an extension of the work done by Hutcherson and Ye (J Micromech Microeng 14:1726–1733, 2004). The accuracy of the present numerical MD approach is verified by comparing its results with the experimental results available in the literature and the finite element method results.

Efficient Numerical Evaluation of Semianalytical Models for Squeeze Film Damping for Torsion Mirrors

Abstract In the semianalytical models for squeeze film damping, the coefficient of damping torque is expressed as an infinite double series. To work with these models, methods for the efficient numerical evaluation of these double series are important, because, as has been pointed out, the results are given by complicated equations; the application of the results is difficult. This paper presents a transformation of the equations that allows a fast and reliable numerical evaluation of the coefficient of damping torque for torsion mirrors. We give precise error bounds and present examples that illustrate that approximations with one or two terms are often sufficient in practice.

Damping and Inertia Coefficients for Two Open Ends Squeeze Film Dampers With a Central Groove: Measurements and Predictions

Aircraft engine rotors are particularly sensitive to rotor imbalance and sudden maneuver loads, since they are always supported on rolling element bearings with little damping. Most engines incorporate squeeze film dampers (SFDs) as means to dissipate mechanical energy from rotor vibrations and to ensure system stability. The paper quantifies experimentally the forced performance of a SFD comprising two parallel film lands separated by a deep central groove. Tests are conducted on two open ends SFDs, both with diameter D = 127 mm and nominal radial clearance c = 0.127 mm. One damper has film lands with length L = 12.7 mm (short length), while the other has 25.4 mm land lengths. The central groove has width L and depth 3/4 L. A light viscosity lubricant flows into the central groove via three orifices, 120 deg apart and then through the film lands to finally exit to ambient. In operation, a static loader pulls the bearing to various eccentric positions and electromagnetic shakers excite the test system with periodic loads to generate whirl orbits of specific amplitudes. A frequency domain method identifies the SFD damping and inertia force coefficients. The long damper generates six times more damping and about three times more added mass than the short length damper. The damping coefficients are sensitive to the static eccentricity (up to ∼ 0.5 c), while showing lesser dependency on the amplitude of whirl motion (up to 0.2 c). On the other hand, inertia coefficients increase mildly with static eccentricity and decrease as the amplitude of whirl motion increases. Cross-coupled force coefficients are insignificant for all imposed operating conditions on either damper. Large dynamic pressures recorded in the central groove demonstrate the groove does not isolate the adjacent squeeze film lands, but contributes to the amplification of the film lands’ reaction forces. Predictions from a novel SFD model that includes flow interactions in the central groove and feed orifices agree well with the test force coefficients for both dampers. The test data and predictions advance current knowledge and demonstrate that SFD-forced performance is tied to the lubricant feed arrangement.

Closed Form Representations of Some Series in Darling’s Model for Squeeze Film Damping with a Rectangular Plate

Compressible squeeze film damping is a phenomenon of great importance for micromachines. For example, for the optimal design of an electrostatically actuated micro-cantilever mass sensor that operates in air, it is essential to have a model for the system behavior that can be evaluated efficiently. An analytical model that is based upon a solution of the linearized Reynolds equation has been given by R.B. Darling. In this paper we explain how some infinite sums that appear in Darling’s model can be evaluated analytically. As an example of applications of these closed form representations, we compute an approximation for the critical frequency where the spring component of the reaction force on the microplate, due to the motion through the air, is equal to a certain given multiple of the damping component. We also show how some double series that appear in the model can be reduced to a single infinite series that can be approximated efficiently.

Analytical and numerical modeling of squeeze-film damping in perforated microstructures

The literature includes a variety of analytical and semi-analytical models to describe squeeze-film damping in MEMS perforated structures. Even if many of them have been validated by means of numerical simulations, nobody seems to have discussed about the accuracy of numerical approaches in this field. In the present paper, we apply both the main analytical models and a commercial finite element software, COMSOL Multiphysics, to solve a good number of squeeze-film problems. They refer to some cases, which were experimentally investigated during the past by different authors. The tested structures are rigid rectangular plates fabricated with different material, different perforation ratio (i.e., the ratio of the hole side to the holes pitch) and different number of perforations. We compare both the analytical and the numerical results with the available experimental data, in order to have an overview about their effectiveness. Numerical simulations offer in all the considered cases valuable agreement with experiments.

A Comparative Study of Thin-Film and Reynolds Equation Simulation Models for Squeeze Film Damping in MEMS Plate Structures

Squeeze film damping effect of MEMS parallel plate structure was analyzed based on thin film and Reynolds Equation in ANSYS under the different Knudsen numbers. Perforation effect of parallel plate with certain size and operating frequency was achieved under the different Knudsen numbers, the simulation results of two methods are very close. For unperforated plate, when Knudsen number is below 0.01, the discrepancy of two simulations is nonsignificant, and it grows up with Knudsen number. But gas rarefaction effects related with Knudsen number was considered in heat transfer analogy theory and used viscosity modification according to Veijola model, two simulations get the same result. For perforated plate, the simulation discrepancy of two methods will be great because of channel flow's effect and also grow up with Knudsen number, it can't be avoided even if the channel flow's effect and viscosity modification were concerned in heat transfer analogy theory.

Frequency-dependent noise analysis and damping in MEMS

The paper presents a combined noise and damping analysis for MEMS structures, based on frequency-dependent behavioral models extracted from finite element simulations. The design of high sensitivity MEMS-based microsystems needs to consider the frequency noise shaping induced by damping phenomena on micro scale motion, for its contribution can be significant in the system level noise analysis. Frequency-dependent behavioral models for squeeze-film damping are generated from finite element analysis simulations. Unlike existing noise analysis reported in literature, based on frequency-independent damping assumption, the paper integrates the damping and noise aspects using the same frequency-dependent models. The results can be used for a noise-based optimization procedure applied to the design of resonating microsensors, as illustrated for the case of a MEMS-based gyroscope.

Tapping Mode Imaging and Measurements with an Inverted Atomic Force Microscope

This report demonstrates the successful use of the inverted atomic force microscope (i-AFM) for tapping mode AFM imaging of cantilever-supported samples. i-AFM is a mode of AFM operation in which a sample supported on a tipless cantilever is imaged by one of many tips in a microfabricated tip array. Tapping mode is an intermittent contact mode whereby the cantilever is oscillated at or near its resonance frequency, and the amplitude and/or phase are used to image the sample. In the process of demonstrating that tapping mode images could be obtained in the i-AFM design, it was observed that the amplitude of the cantilever oscillation decreased markedly as the cantilever and tip array were approached. The source of this damping of the cantilever oscillations was identified to be the well-known "squeeze film damping", and the extent of damping was a direct consequence of the relatively shorter tip heights for the tip arrays, as compared to those of commercially available tapping mode cantilevers with integrated tips. The functional form for the distance dependence of the damping coefficient is in excellent agreement with previously published models for squeeze film damping, and the values for the fitting parameters make physical sense. Although the severe damping reduces the cantilever free amplitude substantially, we found that we were still able to access the low-amplitude regime of oscillation necessary for attractive tapping mode imaging of fragile molecules.

Interaction of Squeeze Film Dampers and Hole Feed Systems and Its Influence on the Dynamics of a Jeffcott Rotor

The oil feeding of a squeeze film damper through holes can be an interesting solution to improve the damper′s energy dissipation capability since it increases the journal′s effective squeezing surface. While groove feeding has been extensively studied, only a few works have addressed the problem of hole fed dampers. The squeeze film damper model may be able to estimate the pressure at the feed holes as well as the flow into the film. A fundamental issue in the modeling of the hole feed system is the choice of the adequate boundary conditions. Existing published models consider the supply pressure to be constant at a specific section of the hydraulic system. The pressures at the oil injection holes are then obtained via local flow balances taking into account a linear flow resistance across the feed conduits. The model developed in the present work is based on the application of the simplified form of the energy equation commonly used in hydraulics, with approximate pressure drop coefficients. As a boundary condition we impose the power supplied to the pump. The nonlinear hydraulics problem is coupled with the squeeze flow and the Newton–Raphson method is employed to solve the set of nonlinear equations. The influence of the hole feed system on the unbalance response and stability of a Jeffcott rotor is analyzed in regards to the number of feed holes and the power supplied to the pump. The feed system behavior is also studied.