Equivalent Linearization of a Squeeze Film Damper

1986 ◽  
Vol 108 (4) ◽  
pp. 434-440 ◽  
Author(s):  
Songqi Chen ◽  
Shengpei Liu
Keyword(s):  
Squeeze Film ◽  
Equivalent Linearization ◽  
Cycle Period ◽  
Mean Square ◽  
Damping Force ◽  
Average Force ◽  
Squeeze Film Damper ◽  
Spring Force ◽  
The Difference ◽  
Mean Square Errors

In this paper, the equivalent linearization of an intershaft squeeze film damper in a two shaft engine system is investigated. The two shaft centers at the damper position are assumed to move in different elliptical offset orbits and at synchronous frequency with the unbalanced rotor (e.g., the high pressure rotor). The nonlinear damper force is resolved into two orthogonal components along the absolute coordinate directions and, in turn, each of these force components is supposed to be equivalent to the sum of an average force, a linear spring force, and a linear damping force in the corresponding direction. By using the method of equivalent linearization by harmonic balance, the six parameters of the equivalent forces, including two average forces, two equivalent spring coefficients, and two equivalent damping coefficients, are expressed analytically by the squeeze film forces and the assumed orbital motion of the two shaft centers at the damper position. The analytical expressions of the squeeze film forces are derived from an approximate solution of the basic Reynolds equation. The results obtained are verified by the method of equivalent linearization by minimum mean square errors. It shows that the six obtained parameters make the mean square errors minimum over a cycle period of motion, the errors being the difference between the equivalent forces and the actual nonlinear forces.

Forced Response of a Squeeze Film Damper and Identification of Force Coefficients From Large Orbital Motions

2004 ◽  
Vol 126 (2) ◽  
pp. 292-300 ◽  
Author(s):  
Luis San Andre´s ◽  
Oscar De Santiago
Keyword(s):  
Excitation Frequency ◽  
Air Entrainment ◽  
Forced Response ◽  
Effective Length ◽  
Single Frequency ◽  
Squeeze Film ◽  
Inertia Force ◽  
Damping Force ◽  
Squeeze Film Damper ◽  
Force Coefficients

Experimentally derived damping and inertia force coefficients from a test squeeze film damper for various dynamic load conditions are reported. Shakers exert single frequency loads and induce circular and elliptical orbits of increasing amplitudes. Measurements of the applied loads, bearing displacements and accelerations permit the identification of force coefficients for operation at three whirl frequencies (40, 50, and 60 Hz) and increasing lubricant temperatures. Measurements of film pressures reveal an early onset of air ingestion. Identified damping force coefficients agree well with predictions based on the short length bearing model only if an effective damper length is used. A published two-phase flow model for air entrainment allows the prediction of the effective damper length, and which ranges from 82% to 78% of the damper physical length as the whirl excitation frequency increases. Justifications for the effective length or reduced (flow) viscosity follow from the small through flow rate, not large enough to offset the dynamic volume changes. The measurements and analysis thus show the pervasiveness of air entrainment, whose effect increases with the amplitude and frequency of the dynamic journal motions. Identified inertia coefficients are approximately twice as large as those derived from classical theory.

Inference Based on α-Cut and Generalized Mean with Fuzzy Tautological Rules

2010 ◽  
Vol 14 (1) ◽  
pp. 76-88 ◽  
Author(s):  
Kiyohiko Uehara ◽  
◽  
Takumi Koyama ◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Systems ◽  
Fuzzy Rules ◽  
Mean Square ◽  
Basic Properties ◽  
Generalized Mean ◽  
Conventional Methods ◽  
The Difference ◽  
Mean Square Errors

Theoretical aspects are provided for inference based on α-cuts and generalized mean (α-GEMII). In order to clarify the basic properties of the inference, fuzzy tautological rules (FTRs) are focused on, which are composed by setting fuzzy sets in consequent parts identical to those in antecedent parts of initially given fuzzy rules. It is mathematically proved that the consequences deduced with FTRs are closer to given facts as the number of FTRs increases. The aspects provided in this paper are appropriate from axiomatic viewpoints and can contribute to interpretability in fuzzy systems constructed with α-GEMII. They are not obtained in conventional methods based on the compositional rule of inference. Simulations are performed by evaluating difference (mean square errors) between given facts and deduced consequences under the condition that convex and symmetric fuzzy sets are given as facts. Their results show that the difference becomes smaller as the number of FTRs increases. Thereby, it is confirmed that α-GEMII has an advantage in the interpretability with respect to FTRs over the conventional methods.

Squeeze Film Damper With a Mechanical End Seal: Experimental Force Coefficients Derived From Circular Centered Orbits

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Luis San Andrés ◽  
Adolfo Delgado
Keyword(s):  
Dry Friction ◽  
Dynamic Pressure ◽  
Test System ◽  
Single Frequency ◽  
Squeeze Film ◽  
Mechanical Seal ◽  
Damping Force ◽  
Squeeze Film Damper ◽  
Force Coefficients ◽  
Aircraft Application

The paper presents parameter identification measurements conducted on a squeeze film damper (SFD) featuring a nonrotating mechanical seal that effectively eliminates lubricant side leakage. The SFD-seal arrangement generates dissipative forces due to viscous and dry-friction effects from the lubricant film and surfaces in contact, respectively. The test damper reproduces an aircraft application that must contain the lubricant for extended periods of time. The test damper journal is 2.54cm in length and 12.7cm in diameter, with a nominal clearance of 0.127mm. The damper feed end opens to a plenum filled with lubricant, and at its discharge grooved section, four orifice ports evacuate the lubricant. In earlier publications, single frequency force excitation tests were conducted, without and with lubricant in the squeeze film land, to determine the seal dry-friction force and viscous damping force coefficients. Presently, further measurements are conducted to identify the test system and SFD force coefficients using two sets of flow restrictor orifice sizes (2.8mm and 1.1mm in diameter). The flow restrictors regulate the discharge flow area and thus control the oil flow through the squeeze film. The experiments also include measurements of dynamic pressures at the squeeze film land and at the discharge groove. The magnitude of dynamic pressure in the squeeze film land is nearly identical for both sets of flow restrictors, and for small orbit radii, dynamic pressures in the discharge groove have peak values similar to those in the squeeze film land. The identified parameters include the test system damping and the individual contributions from the squeeze film, dry friction in the mechanical seal and structure remnant damping. The identified system damping coefficients are frequency and motion amplitude dependent due to the dry-friction interaction at the mechanical seal interface. Squeeze film force coefficients, damping and added mass, are in agreement with simple predictive formulas for an uncavitated lubricant condition and are similar for both flow restrictor sizes. The SFD-mechanical seal arrangement effectively prevents air ingestion and entrapment and generates predicable force coefficients for the range of frequencies tested.

Identification of Force Coefficients in a Squeeze Film Damper With a Mechanical End Seal: Centered Circular Orbit Tests

2007 ◽  
Author(s):  
Luis San Andre´s ◽  
Adolfo Delgado
Keyword(s):  
Dry Friction ◽  
Dynamic Pressure ◽  
Test System ◽  
Single Frequency ◽  
Squeeze Film ◽  
Mechanical Seal ◽  
Damping Force ◽  
Squeeze Film Damper ◽  
Force Coefficients ◽  
Aircraft Application

The paper presents parameter identification measurements conducted on a squeeze film damper (SFD) featuring a non-rotating mechanical seal that effectively eliminates lubricant side leakage. The SFD-seal arrangement generates dissipative forces due to viscous and dry-friction effects from the lubricant film and surfaces in contact, respectively. The test damper reproduces an aircraft application that must contain the lubricant for extended periods of time. The test damper journal is 2.54 cm in length and 12.7 cm in diameter, with a nominal clearance of 0.127 mm. The damper feed end opens to a plenum filled with lubricant, and at its discharge grooved section, four orifice ports evacuate the lubricant. In prior publications (ASME Paper GT2006-90782, IJTC2006-12041), single frequency force excitation tests were conducted, without and with lubricant in the squeeze film land, to determine the seal dry-friction force and viscous damping force coefficients. Presently, further measurements are conducted to identify the test system and SFD force coefficients using two sets of flow restrictor orifice sizes (2.8 mm and 1.1 mm in diameter). The flow restrictors regulate the discharge flow area, and thus control the oil flow through the squeeze film. The experiments also include measurements of dynamic pressures at the squeeze film land and at the discharge groove. The magnitude of dynamic pressure in the squeeze film land is nearly identical for both sets of flow restrictors, and for small orbit radii, dynamic pressures in the discharge groove have peak values similar to those in the squeeze film land. The identified parameters include the test system damping and the individual contributions from the squeeze film, dry friction in the mechanical seal and structure remnant damping. The identified system damping coefficients are frequency and motion amplitude dependent due to the dry friction interaction at the mechanical seal interface. Squeeze film force coefficients, damping and added mass, are in agreement with simple predictive formulas for an uncavitated lubricant condition and are similar for both flow restrictor sizes. The SFD-mechanical seal arrangement effectively prevents air ingestion and entrapment and generates predicable force coefficients for the range of frequencies tested.

Experimental Investigation on the Dynamic Pressure and Force Response of a Partially Sealed Squeeze Film Damper

1991 ◽  
Author(s):  
G. Meng ◽  
L. A. San Andres ◽  
J. M. Vance
Keyword(s):  
Rotational Speed ◽  
Dynamic Pressure ◽  
Tangential Force ◽  
Radial Force ◽  
Squeeze Film ◽  
Damping Force ◽  
Film Pressure ◽  
Squeeze Film Damper ◽  
Fluid Forces ◽  
Supply Pressure

Abstract The influence of rotational speed, oil temperature and supply pressure on the squeeze film pressure and fluid forces is investigated experimentally for a partially sealed squeeze film damper (SFD) test rig executing circular centered orbits. Experimental Tesults show that the sealed damper produces higher damping forces than an open end SFD, though it is more prone to produce oil cavitation. As a result, the peak-to-peak pressures and the tangential force (damping force) decrease with increasing rotational speed; while, the radial force (stiffhening force) becomes negative due to the large extent of the cavitation zone. The tangential force decreases and the radial force increases with increasing lubricant temperature. The squeeze film pressure and film force increase as the supply pressure rises. The film cavitation onset is determined by the level of supply pressure and rotational speed.

Application of the k-ε Turbulence Model to the Squeeze Film Damper

1987 ◽  
Vol 109 (1) ◽  
pp. 164-168 ◽  
Author(s):  
Chiao-Ping Ku ◽  
John A. Tichy
Keyword(s):  
High Speed ◽  
Transport Model ◽  
Tangential Force ◽  
Fluid Pressure ◽  
Radial Force ◽  
Squeeze Film ◽  
Fluid Inertia ◽  
Damping Force ◽  
Mean Velocity ◽  
Squeeze Film Damper

The one-dimensional squeeze film damper is modeled for high speed flow by using the two-equation (k-ε) turbulent transport model. The assumption is made that the fluid flow at each local region of the squeeze film damper has similar behavior to inertialess flow in a channel at comparable Reynolds number. Using the k-ε model, the inertialess channel flow case is solved. Based on this result, correlations are obtained for the mean velocity, inertia and viscous terms of the integrated momentum equation for the squeeze film damper. It is found that turbulence increases the magnitude of the fluid pressure and the tangential force, while fluid inertia causes a shift on the pressure creating a significant radial force. In applications, turbulence may be a beneficial effect, increasing the principal damping force; while inertia may be detrimental increasing the cross-coupling forces.

The Loss of Dexterity in the Bilateral Lower Extremities in Patients With Stroke

2011 ◽  
Vol 27 (2) ◽  
pp. 122-129 ◽  
Author(s):  
Ryoji Kiyama ◽  
Kiyohiro Fukudome ◽  
Toshiki Hiyoshi ◽  
Akihide Umemoto ◽  
Yoichi Yoshimoto ◽  
...  
Keyword(s):  
Root Mean Square Error ◽  
Mean Square Error ◽  
Root Mean Square ◽  
Knee Extensor ◽  
Lower Extremities ◽  
Response Force ◽  
Mean Square ◽  
Control Subjects ◽  
The Difference ◽  
Mean Square Errors

The aim of this study was to examine the dexterity of both lower extremities in patients with stroke. Twenty patients with stroke and 20 age-matched control subjects participated in this study. To determine the dexterity of the lower extremities, we examined the ability to control muscle force during submaximal contractions in the knee extensor muscles using a force tracking task. The root mean square errors were calculated from the difference between the target and response force. The root mean square error was significantly greater in the affected limb of patients with stroke compared with those of the unaffected limb and the control subjects, and in the unaffected limb compared with that of the control subjects. Furthermore, the root mean square error of the affected limb was related significantly to motor function as determined by Fugl-Myer assessment. These results demonstrate impairment of the dexterity of both the affected and the unaffected lower extremities in patients with stroke.

Squeeze-Film Phenomena in Microresonators Dynamics: A Mathematical Modeling Point of View

2006 ◽  
Author(s):  
G. Nakhaie Jazar ◽  
M. Mahinfalah ◽  
A. Khazaei ◽  
M. H. Alimi ◽  
J. Christopherson
Keyword(s):  
Mathematical Modeling ◽  
Fluid Layer ◽  
Dynamic Properties ◽  
Contact Forces ◽  
Squeeze Film ◽  
Design Parameters ◽  
Damping Force ◽  
Engine Mounts ◽  
Squeeze Film Damping ◽  
Spring Force

Oscillating microplates attached to microbeams is the main part of many microresonators. There are several body and contact forces affecting a vibrating microbeam. Among them are some forces appearing to be significant in micro and nano size scales. Accepting an analytical approach, we present the mathematical modeling of a microresonator a nonlinear model for MEMS are presented, which accounts for the initial deflection due to polarization voltage, mid-plane stretching, and axial loads as well as the nonlinear displacement coupling of electric force. The equations are nondimensionalized and the design parameters are developed. However, the main purpose of this investigation is to present an applied model to simulate the squeeze-film phenomena. We separate the two characteristics of the squeeze-film phenomena and model the damping and stiffness effects individually. Motion of the microplate and flow of the gas underneath is similar to the function of a decoupler plate in hydraulic engine mounts (Golnaraghi and Nakhaie Jazar 2001, Nakhaie Jazar and Golnaraghi 2002). More specifically, the squeeze-film damping is qualitatively similar to the function of decoupler plate in hydraulic engine mounts, which is an amplitude dependent damping (Christopherson and Nakhaie Jazar 2005). It means squeeze-film damping effect is a positive phenomenon to isolate the vibration of microplate from the substrate. Following Golnaraghi and Nakhaie Jazar (2001), and utilizing the aforementioned similarity, we model the squeeze-film damping force, fsd, by a cubic function, where the coefficient Cs is assumed constant and must be evaluated experimentally. In the simplest case, we present the following fifth degree function to simulate the spring force, fss, of the squeeze film phenomenon, simply because at low amplitudes, w ≈ 0, the fluid layer is not strongly squeezed and there is no considerable resistance. On the other hand, at high amplitudes, w ≈ d, there is not much fluid to react as a spring. In addition, speed is proportionally related to the squeezeness of trapped fluid. The coefficient ks assumed constant and must be evaluated experimentally. The coefficients cs and ks are dependent on geometry as well as dynamic properties of the fluid, but assumed to be independent of kinematics of the microbeam such as displacement and velocity. Therefore, this investigation presents two mathematical functions to describe stiffness and damping characteristics of squeeze-film phenomena in a reduced-order model of microresonators.

Application of the Magnetorheological Squeeze Film Dampers for Reducing Energy Losses in Supports of Rotors of Rotating Machines

2017 ◽  
Author(s):  
Jaroslav Zapoměl ◽  
Petr Ferfecki
Keyword(s):  
Mathematical Model ◽  
Dynamical Analysis ◽  
Squeeze Film ◽  
Energy Losses ◽  
Damping Force ◽  
Squeeze Film Damper ◽  
Damping Effect ◽  
Wide Range ◽  
Squeeze Film Dampers ◽  
Computational Procedures

Unbalance of rotating parts is the main source of excitation of lateral oscillations of rotors, of increase of time varying forces transmitted to the rotor stationary part, and of energy losses generated in the support elements. The technological solution, which makes it possible to reduce these undesirable effects, consists in adding damping devices to the rotor supports. A simple dynamical analysis shows that to achieve their optimum performance their damping effect must be adaptable to the current operating speed. This is enabled by magnetorheological squeeze film dampers, the damping effect of which is controlled by the change of magnetic flux passing through the lubricating layer. The developed mathematical model of the magnetorheological squeeze film damper is based on assumptions of the classical theory of lubrication and on representing the magnetorheological oil by a bilinear material. The results of the carried out computational simulations show that the appropriate control of the damping force makes it possible to minimize the energy losses in a wide range of operating speeds. The development of a new mathematical model of the magnetorheological squeeze film damper, the extension of computational procedures, in which this model has been implemented, the confirmation that the magnetorheological dampers make it possible to reduce energy losses in the rotor supports, and learning more on influence of controllable dampers on behavior of rotor systems are the principal contributions of the presented paper. The carried out research highlights the possibility of reducing the energy losses by means of employing magnetorheological squeeze film dampers, which represents a new field of their prospective application.

scholarly journals Dynamic Analysis of Squeeze Film Damper Supported Rotors Using Equivalent Linearization

1993 ◽  
Author(s):  
A. El-Shafei ◽  
R. V. Eranki
Keyword(s):  
Dynamic Analysis ◽  
Numerical Integration ◽  
Dynamic Characteristics ◽  
Nonlinear Dynamic Analysis ◽  
Least Square ◽  
Squeeze Film ◽  
Equivalent Linearization ◽  
Squeeze Film Damper ◽  
Elliptic Orbits ◽  
Rotor Dynamic

The technique of equivalent linearization is presented in this paper as a powerful technique to perform nonlinear dynamic analysis of squeeze film damper (SFD) supported rotors using linear rotordynamic methods. Historically, it is customary to design squeeze film dampers (SFDs) for rotordynamic analysis by assuming circular centered orbits, which is convenient in making the nonlinear damper coefficients time independent and thus can be used in an iterative approach to determine the rotor dynamic characteristics. However, the general synchronous orbit is elliptic in nature due to possible asymmetry in the rotor support. This renders the nonlinear damper coefficients time dependent which would require extensive numerical computation using numerical integration to determine the rotor dynamic characteristics. Alternatively, it is shown that the equivalent linearization, which is based on a least square squares approach, can be used to obtain time independent damper coefficients for SFDs executing eccentric elliptic orbits which are nonlinear in the orbit parameters. The resulting equivalent linear forces are then used in an iterative procedure to obtain the unbalance response of a rigid rotor-SFD system. Huge savings over numerical integration are reported for this simple rotor. The technique can be extended to be used in conjunction with currently available linear rotordynamic programs to determine the rotor dynamic characteristics through iteration. It is expected that for multi-rotor multi-bearing systems this technique will result in even more economical computation.

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