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# MECHANICAL VIBRATION MEEN 617

Texas A&M

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This 21 page Class Notes was uploaded by Orrin Weissnat on Wednesday October 21, 2015. The Class Notes belongs to MEEN 617 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/225994/meen-617-texas-a-m-university in Mechanical Engineering at Texas A&M University.

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it ll cwl39wlvtr gill lii l iilHllE quot l Ei T quotllJill lull lH39lLlle n lL Motion ie time varying changes is ubiquitous in nature All systems small and large simple or complex describing natural physical biological or social phenomena are subject to change and variation A system is an ensemble of components acting as a whole In the case of a mechanical system this is described by relationships of energy transfer between its components and also with its surroundings The system components are designed to satisfy a technical goal ie convert one type of energy to another such as in a steam turbine an electric motor a compressor etc The basic mechanical components are shafts bearings wheels couplings gears belts chains cams structural support elements piping etc The study ofthe dynamic response of a system includes Design Conceptualize system of interest describe its functions and separate it from others Analysis Create a model that defines as closely as possible the nature of the relationships between its parts and determine the dynamic response to a set of realistic conditions Testing Measurement of the dynamic response on a real system or prototype to confirm analytical predictions MEEN 617 Notes Introduction LuisSan Andr s 2008 1 STEPS in Modeling a Mechanical System The steps to follow in the analysis of mechanical systems are a Establish the necessary assumptions provide pictorial representations free body diagrams determine similaritiesdifference from other systems etc ie bring the REAL PROBLEM into an ANALYTICAL MODEL b MATHEMATICAL MODEL identify constraints amp establish degrees of freedom apply the fundamental principles of motion to the analytical model and derive equations of motion governing the response of the system Give attention to initial conditions and external forcing functions The principles used are 9 Conservation of linear momentum and angular momentum or 9 Conservation of mechanical energy c Find the dynamic behavior ie solve the mathematical model within the range of parameters of interest and determine the goodness or badness of proposed design A more detailed systemic approach is given by 1 Define the system and its fundamental components 2 List all assumptions and constraints 3 Select significant input excitations and outputs responses 4 Model the system components constitutive equations 5 Model the system define relationships between components 6 Solve for required responses given a set of known excitations for a system configuration 7 Check for consistency of results no violation of item 2 8 Interpret the system response 9 Provide recommendations design changes and conclusions MEEN 617 Notes Introduction LuisSan Andr s 2008 THE STUDY OF THE DYNAMIC RESPONSE OF MECHANICAL SYSTEMS The mathematical models of mechanical systems are of two classes 1 Continuous Models represented by an infinite number of degrees of freedom and usually described by partial differential equations and 2 DiscreteParameter or LumpedMass Models represented by a finite number of degrees of freedom and described by ordinary differential equations as given above What model is the most adeguate to select This is determined by the type of behavior the system is expected to show or desired to perform forthe conditions of interest Simplicity is most desirable but model must always replicate the physics of the system In the study of mechanical systems we will concentrate on the dynamics of lumped parameter linear systems These systems are deterministic and where the principle of superposition holds In a system there is a specified set of dynamic variables called lNPUTs or excitations and a dependent set called OUTPUTs or responses For nonlinear systems whose behavior depends greatly on their initial state we will generally consider small amplitude motions or changes about an equilibrium position This assumption brings most often linearity into the dynamics ofthe system of interest We will learn that lumpedparameter mechanical systems undergoing oscillatory motions vibrations can be modeled as Second Order Systems described by ordinary differential equations ODEs and initial conditions at time t 0 XZO X0 XZO X V 0 0 where M D K are generalized inertia damping and stiffness elements describing the system and Xt and Ft are generalized time dependent displacement and external forces respectively MEEN 617 Notes Introduction LuisSan Andr s 2008 3 We are interested in the study of the dynamic response the time dependent changes of a mechanical vibratory system due primarily to two types of considerations a Free Response Motion resulting from specified initial conditions disturbances to an equilibrium or steadystate configuration b Forced Response Motion resulting from specified external inputs or load excitations to the system In the last decade high speed computers and advanced numerical modeling techniques have helped in understanding the dynamic response of complex systems with large number of degrees of freedom However you need to always remember that a computer is iust a tool to solve not to understand a problem In MEEN 617 you will tackle the analysis of systems not by computer You will be successful if you can are able to master 0 knowledge of the physical laws of motion identification of the fundamental parameters or elements describing the system understanding of the equations of motion governing the system behavior and understanding the beauty of simple yet accurate solutions valid for systems of any complexity We will devote considerable time to obtain the dynamic response solution ofthe equations above for many different mechanical systems To this end we will learn the necessary analytical tools to derive the equations above and also the effective means to obtain solutions predicting the dynamic response of a vibratory system MEEN 617 Notes Introduction LuisSan Andre39s 2008 4 Independent of the physical details of the system considered the analysis of the dynamic response of a vibratory mechanical system should aid to answer the following fundamental questions gtHow does the system respond with time for any particular type of disturbance gtHow long will it take for the dynamic action to dissipate if the disturbance is briefly applied and then removed Whether the system is stable or if its oscillations will increase in magnitude with time even after the disturbance has been removed gtWhat modifications can be made to the system to improve its dynamic characteristics with regard to some specific application That is the ultimate purpose of the modelinganalysis is to answer relevant questions about the design amp performance of a mechanical system such as Will the system operate Will it have the rated design performance Will operation be stable static and dynamic Does it meet vibration characteristics Will any part break under normal operation Will it be reliable With life how long What are its operating limits And for how long can operate safely Why does partheep breaking while system operates ls modification Yable to improve performance MEEN 617 Notes Introduction LuisSan Andr s 2008 Application example Experimental identi cation of bearing force coef cients Consider a test bearing or seal element as a point mass undergoing forced Vibrations induced by external forcing functions The equations of motion for small amplitudes about an equilibrium position are described in linear form as 1 Km CXX 7 KhYY ChYY FY Representation of point mass and bearing force coefficients used for identification of parameters from dynamic load and motion measurements CM KXYJ7 1 CYXX l Krr KhY KYXX fr Notes 15 Identification of bearing force coefficients Dr Luis San Andres 2008 l Where gXY are external excitation forces Mh is the test element mass KhlChiiXY are structural support stiffness and damping coefficients and KyCUhFXY are the sealbearing dynamic stiffness and damping force coefficients respectively Inertia force coefficients are not accounted for in the model above These coefficients are insignificant for highly compressible uids LH2 or air and for most bearing applications with mineral lubricants This apparent simplification is easily removed and does not diminish the importance of the identification method The structural stiffness and damping coefficients KhChXy are obtained from prior shake results under dry conditions ie Without uid through the test element Two independent forced excitations impact periodicsingle frequency sineswept random etc fX0T and 0 T for example are applied to the test element This process can be written as f X1 x10 1 Apply and measure 2a fyl I yla X2 x20 2 Apply and measure 2b y2 I yzo 3 Obtain the discrete Fourier transform FFT of the applied forces and displacements ie Let FX FFT ff X FFTFK FY10 qut 7 Yum ylm 2b FXM FFT fxzm X200 FFTx2z FY2m fY2t Yzm yZU Notes 15 Identification of bearing force coefficients Dr Luis San Andres 2008 2 The FFT is an operation that transforms the information from the time domain into the frequency domain Incidentally recall that m XW FFTx a9 Xm FFTx 2c 4 For the assumed physical model the equations of motion in the frequency domain become Mha2X1CXX ChXz aX1 CXYz aY1 KXXKILXX1KXYYTFX1 2d Ma2 Y1 CW CWin CYXiaX1 2 KYYKhYY1KYXX1FY1 6 Or written in matrix form as Define complex impedances1 HZZXY as 4 where i V l 61 I forz39 j X Y zero otherwise The impedances are composed of real and imaginary parts both functions of frequency 0 The real part denotes the dynamic stiffness while the imaginary part 1 This is as you know a misnomer Dynamic complex stiffness is a more appropriate name Notes 15 Identification of bearing force coefficients Dr Luis San Andres 2008 3 quadrature stiffness is proportional to the viscous damping coefficient as shown in the figure below 8 E a E E XX 1 E q M 6 5 10 0 200 400 600 800 1000 frequency rads ReH ImH Real and imaginary parts of ideal mechanical impedance With definition 4 the EOMs 3 become for the first measurement 1 32 quotam I V1923 l lt5 1 rm JE W 39Lam tbn 1L 517 s when Add these two equations gives and reorganize them as Robison et al 1995 Notes 15 Identification of bearing force coefficients Dr Luis San Andres 2008 4 Equation reresents four independent equations with four unknowns easily found using Cramers rule for example or The meaning of linear independence of the test forces and ensuing motions should now be clear That is the forces in the second test cannot just be a multiple of the first set of forces since then the matrices of displacements and forces would be singular IN general the experimenter has to chose sets of excitations that are linearly independent for example fX0T and 0 T are preferred choices Preliminary estimates of the system parameters 7 2 I determined by curve fitting of discrete impedances HZj to the test data over a preselected frequency range Rouvas and Childs 1993 use this impedance identification method exclusively for identification of force coefficients in hydrostatic bearings and seals with water as the lubricant System transfer functions outputinput could be used to obtain more precise estimates of the sealbearing force coefficients Nordmann and Schollhorn 1980 Massmann and Nordmann 1985 In terms of the impedances Hy1UjXy the transfer functions describing system exibilities are generated by the following equations Notes 15 Identification of bearing force coefficients Dr Luis San Andres 2008 5 H H GXXTFX1 AW GXYZTFX2 AXY H H GYX TFY1 AYX GYY TFYz XX 7 where A HXXHYY HXYHYX Next the Instrumental Variable IV method of Fritzen 1985 an extension of a leastsquares estimation method is used to simultaneously curve fit all four transfer functions from measurements in two orthoonal directions V in quotm m gut with J wnrjiu im identically equal to the identity matrix since However in any measurement process there is some noise associated with the experimental measurements Thus an error matrix N is introduced into the relationship GHGK a2MiaCIN 8 where K M and C are the matrices of system stiffness mass and damping coefficients Rearranged this equation becomes A M IN C 9 Notes 15 Identification of bearing force coefficients Dr Luis San Andres 2008 6 where A contains the measured transfer functions Solution of Eq 9 by leastsquares requires minimization of the loss function defined by the Euclidean norm of N This minimization leads to the normal equations 10 A first set of force coefficients is determined fromthese equations Using the IV method extenion the weighting function quot is replaced by a new matrix function created from analytical transfer functions resulting from the initial leastsquares curve fit This weighting function is free of measurement noise and contains a peak only at the resonant frequency as determined from the first estimates of stiffness mass and damping coefficients The calculation cycle is continued until correlation is within a desired tolerance Ransom 1997 Note that the stiffness and damping coefficients are identified in the frequency domain Thus magnitudes of uncertainty for the estimated force coefficients must be obtained by comparing the original frequency responses with the frequency response of a reference excitation force and associated displacement time response Evaluation of coherence functions then becomes necessary to reproduce the exact variability of the identified force coefficients Ransom 1997 describes at length the frequency domain uncertainty analysis implemented Diaz and San Andres 1999 provide in full a description of the identification method along with a MATHCAD program which allows the fast estimation of system parameters in real time Notes 15 Identification of bearing force coefficients Dr Luis San Andres 2008 7 Closure Read the paper of Diaz and San Andres 1999 following pages for further insight on experimental methods and identification procedures in bearings and seals A MATHCAD program is available for your selfstudy and further learning References Diaz S and L San Andres 1999 quotA Method for Identification of Bearing Force Coefficients and its Application to a Squeeze Film Damper with a Bubbly Lubricant STLE Tribology Transactions Vol 42 4 pp 739746 STLE Paper 99 AM5 Fritzen C P 1985 Identification of Mass Damping and Stiffness Matrices of Mechanical Systems ASME Paper 85DET91 Massmann H and R Nordmann 1985 Some New Results Concerning the Dynamic Behavior of Annular Turbulent Seals Rotordynamic Instability Problems of High Performance Turbomachineiy Proceedings of a workshop held at Texas AampM University Dec pp 179194 Nordmann R and K Schollhorn 1980 Identification of Stiffness and Damping Coefficients of Journal Bearings by Means of the Impact Method Proceedings of the 2nd International Conference on Vibrations in Rot Mach IMechE pp23123 8 Ransom D L 1997 Identification of Dynamic Force Coefficients of a Labyrinth and Gas Damper Seals Using Impact Load Excitations Master s Thesis Texas AampM University December Robison M G Arauz and L San Andres 1995 A Test Rig for the Identification of Rotordynamic Force Coefficients of Fluid Film Bearing Elements ASIVIE Paper 95GT431 Rouvas C and D Childs 1993 A Parameter Identification Method for the Rotordynamic Coefficients of a High Speed Reynolds Number Hydrostatic Bearing ASIVIE Journal of Vibration and Acoustics Vol 115 pp 264270 Notes 15 Identification of bearing force coefficients Dr Luis San Andres 2008 8 Vol 42 1999 4 739746 TRIBOLOGY TRANSACTIONS A Method for Identification of Bearing Force Coefficients and Its Application to a Squeeze Film Damper with a Bubbly Lubricant SERGIO E DIAZ and LUIS A SAN ANDRES Member STLE Texas AampM University Mechanical Engineering Department College Station Texas 77843 A general formulation of the instrumental variable lter IVF method for parameter identification of a nDOF Degrees Of Freedom mechanical linear system is presented The I VF is afre quency domain method and an iterative variation of the least squares approximation to the system exibilities Weight functions constructed with the estimated exibilities are introduced to reduce the e ect of noise in the measurements thus improving the estimation of dynamic force coe icients The l VF method is applied in conjunction to impact force excitations to estimate the mass stiffness and damping coe icients of a test rotor supported on a squeeze lm damper SFD operating with a bubbly lubri cant The amount ofair in the lubricant is variedfrom nil to 100 Presented at the 54th Annual Meeting Las Vegas Nevada May 2327 1999 Final manuscript approved March 24 1999 percent to simulate increasing degrees of severity of air entrain ment into the damper lm lands The experimental results and parameter estimation technique show that the SF D damping force coe icients increase as the air volume fraction in the mixture increases to about 50 percent in volume content The damping coef cients decrease rapidly for mixtures with larger air concen trations The unexpected increase in direct damping coe icients indicates the complexity of the SFD bubbly ow eld and war rants mher experimental veri cation KEY WORDS Squeeze im Lubrication Dampers Bearings INTRODUCTION Experimental identification of linearized bearing parameters namely stiffness and damping force coef cients is of importance to verify the rotordynamic performance of actual uid lm bear NOMENCLATURE A matrix of coefficients for error equation 2nN3n c SFD nominal radial clearance 0290 mm C matrix of damping coef cients nxn Ci equivalent damping coef cients of SFDrotor system Nsecm shaft diameter 95 mm joumal diameter 508 mm error matrix nn extended error matrix 2nNn forcing vector n DFT of the force vector f n flexibility matrix nn impedance matrix nn imaginary unit ll identity matrix nn extended identity matrix 2nNn indexes for degrees of freedom l2n quot Irma mul tDm L 739 frequency index matrix of stiffness coef cients nn equivalent stiffness coef cients of SFDrotor system Nm shaft length 3048 mm joumal length 254 mm iteration counter for IVF method matrix of inertia coef cients nn equivalent inertia coef cients of SFD rotor system kg number of degrees of freedom of the system number of frequencies considered for identi cation range time sec weight matrix for IV method 2nN3n displacements state vector n DEF of the displacement vector x n PNgt 335 gtltgtltgtlt 23 lt horizontal and vertical coordinates respectively fluid viscosity Pas frequency rads 08 zero null matrix nn 740 S DIAZ AND L SAN ANDRES ing elements and to validate and calibrate predictive tools for computation of bearing and seal dynamic forced responses The ultimate goal is to provide reliable data bases from which to deter minc the con dence of bearing andor seal operation under both normal design conditions and extreme environments due to unforeseen events In addition even advanced analytical models are very limited or nonexisting for certain bearing and seal con gurations and with stringent particular operating conditions and thus experimental measurements of actual bearing force coef cients constitute the only option available to generate engineering results of interest Squeeze film dampers operating with air entrainment are but an example of the many applications where systematic experimentation becomes mandatory The estimation of bearing and seal rotordynamic force coef cients has been traditionally based on time domain response pro cedures I However these techniques are limited in their scope use only a limited amount of the recorded information and often provide poor results with marginal con dence levels 2 Modern bearing parameter identification techniques are based on frequen cy domain procedures where dynamic force coef cients are esti mated from transfer functions of measured displacements veloc ities and accelerations as well due to external loads of a pre scribed time varying structure The frequency domain methods takc advantage of high speed computing and processors thus pro ducing estimates of system parameters in real time and at a frac tion of the cost and effort prevalent with cumbersome time domain techniques 35 This paper presents a frequency domain method for identifica tion of linearized bearing force coef cients from test uid lm bearing elements The technique a variation of a least square esti mator is based on the Instrumental Variable Filter IVF Method with the capability to automatically reduce the noise inherent in any measurement and to provide reliable bearing force coef cients within a frequency range The analysis introduces the equa tions of motion for the test system and the measurement of time domain responses The description follows with the transforma tion of displacement and load dynamic responses to the frequency domain and the implementation of the procedure for error mini mization and curve tting of the outputinput transfer functions over a selected frequency range The identification method is applied to the estimation of sys tem force coefficients K0 CU MU 1le for a small test rotor sup ported on a squeeze lm damper SFD Calibrated impact guns excite the rotor in two radial planes X Y and the rotor displace ments are recorded for a multiple sequence of impacts The SFD operates with an air in oil bubbly mixture to simulate prevalent operating conditions with air entrainment 6 The identi cation procedure also renders dry ie without lubricant structural force coef cients INSTRUMENTAL VARIABLE PARAMETER IDENTIFICATION TECHNIQUE Consider a ndegree of freedom linear mechanical system gov erned by the following system of differential equations MX CX KX39 41 where fm and x0 represent the external forcing function and sys tem displacements respectively The 39 denotes differentiation with respect to time The square matrices M K and C contain the generalized mass stiffness and damping force coef cients repre senting the parameters of the system The objective of the identi cation procedure is to determine the system force coef cients from measurements of the system dynamic response due to applied external loads The governing equations can be written in the frequency domain as sz quotC K th tmxtw Ftw 2 where Xw and F m are the discrete Fourier transforms DFT of the time varying forces and displacements f and x ly The impedance of the system is generally de ned as respective m a2MiaC K 1 J T 3 The n2 impedance coef cients i39iiiln are complex algebra ic functions of the excitation frequency 0 However the system of Eq 2 provides n equations for n2 unknowns A nDOF Degrees Of Freedom system has nlinearly independent modes of vibration Thus nlinearly independent excitations lgHM should lead to nIinearly independent responses XiHm hence rendering nlinearly independent systems of equations of the form Eq 2 for any given excitation frequency The selection of the set of force excitations depends fundamentally on the structure and constraints of the system A typical method consists of exciting the system at the location of each degree of freedom one at the time Note that this procedure when canned out with static loads leads naturally to the determination of the system exibilities ie the in uence coef cient method However any combination of forc ing functions is appropriate as long as the nforces are linearly independent The nsystems of Eq 2 representing the independent meas urements can be regrouped in the following form I 39 2 39 39 n l 39 2 39 39 n anlxuu Xco L XmllFw Fw L l w 4 and the system impedance coef cients at the frequency of interest can be computed from I 39 2 39 39 n I 39 2 39 39 n 1 u lFtwll Ftw L FtwlXw Xltwgt L thl 5 The de nition of the impedance coef cients Eq 3 renders a quadratic relationship in frequency To identify the force coef cients it is suf cient in principle to obtain the impedance coef cients at three different and well spaced frequencies and then use some curve t procedure to extract the force coef cients M i J C KiJiJ392Vln Note that the model assumes the force coef cients or system parameters are constants independent of frequency In the following the basic issues related to the selection of appropriate A Method for Identi cation of Bearing Force Coef cients and Its Application to a Squeeze Film Damper with a Bubbly Lubricant 741 test frequencies are discussed In a linear system excited by a sustained time varying force the system response has the same frequency content as the exter nal excitation as long as the transient motions not due to the exter nal force have died out Therefore in an ideal case only puretone forced excitations are required and response measurements con ducted at only three different excitation frequencies should be suf ficient to fully determine the system physical parameters In prac tice however measurements of forces and displacements contain noise that affects greatly the desired results In some other cases the objective is to find linearized force coef cients that represent the behavior of a certain nonlinear system over a frequency range In both circumstances whether dealing with measurement noise or localized system linearization the identi cation proce dure leads to a problem where the minimization of errors is of importance Instead of working with the minimum amount of frequencies needed it is best to obtain measurements for a whole set of fre quencies within a range of interest However an increased cost and time in the experimental procedure is the natural conse quence if the measurements are conducted with a pure tone force excitation for every frequency of interest Therefore other forms of force excitations must be sought The two excitations most commonly used are the impact load and the multiharmonic force though the sweep sine force is also often employed 7 The fundamental idea is to excite the system with a wideband spectrum force which will result in a wideband system frequency response The application of the DFT to the measured forces and displacements leads to discrete algebraic equations in the fre quency domain and at the selected say N frequencies within the range of interest The km impedance coef cients at the frequency wk could then be found from auntie sadlxza a at 6 From here on several paths could be followed to determine the 3n2 parameters MU CU Ki39jiJL2wn from the n2 impedance coef cients lldlwtm as functions of the excitation frequency The most direct and most commonly used procedure consists of per forming independent leastsquares curve fittings to the real and imaginary pants of each component of the impedance matrix over a range of frequencies This procedure takes advantage of the fact that each system coef cient MU CU Kw appears only in one impedance term making the polynomial curve t quadratic for the real part and linear for the imaginary part independent of each other However the direct leastsquares curve fit of the system impedances is highly sensitive to the level ofthe inherent noise in the measurements and to the selection of the frequency range for the approximation 8 A more robust method is achieved based on the following identity 4 k k F wk E 7 where 5 represents the measured exibility matrix de ned as the inverse of the impedance k Eq 6 at the frequency 0k 1 in the equation above corresponds to the estimated system impedance as de ned by Eq 3 E is the matrix of errors due to the approxi mation In this formulation the exibility coef cients work as weight functions of the errors in the minimization procedure Whenever the exibility coef cients are large the error is also penalized by a larger value As a result the minimization proce dure will become better in the neighborhood of the system reso nances natural frequencies where the dynamic exibilities are maximums ie null dynamic stiffness Kwz M That is the measurements containing resonance regions will have more weight on the tted system parameters This result is of impor tance since forcing functions exciting the system resonances are more reliable since this is more sensitive at those frequencies and the measurements are accomplished with larger signal to noise ratios In addition it is precisely around the resonant frequencies where all the physical parameters mass damping and stiffness most affect appreciably the system response For too lowquot fre quencies the important parameter is the stiffness while for too high frequencies the inertia dominates the response Only near the resonance do allithree parameters have an important effect on the system response Therefore it is more convenient to minimize the approximation errors using Eq 7 rather than directly curve tting the impedances However this last procedure could be rather intricate The approximation functions on the lefthandside of Eq 7 are no longer independent of each other since all the parameters appear in all of them This dif culty39 is easily over come by rearranging the impedance de nition Eq 3 to the form M gm a21 icul I E 3915 8 Substituting the de nition Eq 8 into Eq 7 and separating into real and imaginary parts gives Repkw 1rwk11 1 Revat ialrl li fiffoff 39K l0 hall37 9 Stacking the equations for the N discrete frequencies at which the identification procedure is to be performed renders IE i gtltllm l 0 where 742 S DIAZ AND L SAN ANDRES Rel739 alzl raw 1 I 595 Fr39imm lal ImiE l A i l5Nl Zzti i tf l 39 l E Kelli Imile w l le 1 0 1MPquot Fritzen 3 introduces the elegant Instrumental Variable Filter Method lVF to compute the system coefficients that minimize the Euclidean L2 norm of the global error matrix This proce dure was originally developed to estimate parameters in econom etry problems Massmann and Nordmann 4 have applied the method to uid film seal elements The IFV method proposes a solution of the form ml M wmrij iwq Kim 11 The weight matrix W is chosen to have the same form as A see Eq IO but it consists of the analytical exibilities rather than the measured ones ie Re11 w21 01 1 14522 l w21 iwll 1 Re ZN w2vl39 inl ID 01562 1051 inI ID will 12 where M m l 53 wzl fa I K 13 A rst iteration ml is performed with W A which corre sponds to theistandard leastsquares solution of the problem in Eq l0 Then Eq I I is applied iteratively until a given convergence criterion is satis ed This criterion can be conveniently chosen depending on the desired results For example the square sum mation of the differences between the parameters at iteration m and ml can be required to be less than a certain value ie lim iting the Euclidean norm of the error Alternatively it can be required that the largest difference be less than the largest accept able error ie limiting the L norm of the error Different toler ances to each variable could also be asserted depending on their physical units and significance It is clear that the substitution of W for the discrete measured exibility A which also contains noise improves the prediction of the system parameters Note that the product ATA ampli es the noisy components and adds them Therefore even if the noise has a zero mean value the addition of its squares becomes positive resulting in a bias error On the other hand W does not have components correlated to the measurement noise That is no bias error is kept in the product WTA Consequently the approximation to the system parameters is improved An example of the application of the WP parameter identi ca tion method to a simple laboratory rotorbearing system follows Ransom et al 9 provides a successful application for the identi cation of force coef cients in multiplepocket gas damper seals SFDS AND AIR ENTRAINMENT Squeeze lm dampers SFDS are effective means to introduce damping to rotorbearing systems thus reducing vibration ampli tudes at critical speeds and improving system stability A SFD is a type of hydrodynamic bearing in which a nonrotating journal whirls with the shaft and squeezes a thin lm of lubricant that sur rounds it The squeezing action generates hydrodynamic pressures yielding a force that opposes the journal motion and provides the desired damping Generally SFDS operate with low levels of external pressurization and are open to ambient on the sides Under these conditions the cyclic squeezing in and out of the oil results in the entrapment of external air and leads to the formation ofa bubbly foamlike mixture of air and oil within the lm 10 II The mixture has different material properties than the pure lubricant and consequently it affects considerably the dynamic force performance of the SFD Zeidan et al I2 estimate damp ing coef cient losses as large as 75 percent of the value predicted for operation with pure oil The phenomenon of air entrainment is readily acknowledged to be the main obstacle for the reliable prediction of SFD dynam ic forces 13 Yet no accurate measurements correlating the vis cous damping coef cients to the amount of entrained air are avail able The lack of rm quanti able experimental evidence pre vents further advances in the theoretical formulation of SFD flows 14 15 EXPERIMENTAL FACILITY Figure 1 shows a section of the test rig and the instrumentation setup for force and displacements measurement The shaft of length 305 mm 12 and diameter 95 mm 38 is supported by a bronze bushing at the drive end and by a squeeze lm damper at the rotor midspan The squeeze lm damper consists of a steel journal of diameter D and length L equal to 508 mm and 254 mm respectively and a Plexiglas transparent housing The damper radial clearance c is 029 mm 114 mils Four exible rods compose the squirrel cage that supports the damper journal A ball bearing inside the SFDjournal forCes the shaft and the jour nal to whirl together while allowing the shaft to rotate A exible coupling transmits torque from the DC drive motor but isolates lateral vibration A massive disk is mounted on the free end of the shaft to provide inertia and a location to install imbalance masses Two eddy current proximity sensors measuring horizontal and vertical shaft displacements are installed at L 213 mm and LY 254 mm from the rotor drive end respectively The SFD and disk centers are located at LSFD 15 mm and LD 274 mm from the A Method for Identification of Beating Force Coefficients and Its Application to a Squeeze Film Damper with a Bubbly Lubricant data acquisition system motor quot squirrel displacement cage probes exible coupling Fig i Test rig section and instrumentation rotor drive end respectively The bushing stiffness is larger than the SFD elastic support stiffness and thus the rotor pivots about the bushing location for rotor speeds below 6000 rpm shown in Fig 2 For the range of frequencies of interest the rotor can be considered as an equivalent point mass system with two degrees of freedom in the lateral directions X Y A controlled mixture ufair and lSO VG 2 oil flows to the SFD through a small hole located at the top of the bearing housing The viscosity it of the pure lubricant is 225 centipoisc at a tempera ture of 30 C The lubricant exits the test section through both sides of the damper which are open to ambient The mixture is generated in a sparger element installed at the connection of the air and oil lines The proportions of air and oil are accurately reg ulated with valves on each feed line The air volume fraction is computed as the ratio of measured air volumetric flow rate to total air oil volumetric ow rate An instrumented impact gun excites the rotor shaft at the loca tion of the SFD A support allows installation of the impact gun for excitations in the horizontal and vertical directions An AID board and computer record the time traces of the impact force and the shaft lateral displacements simultaneously at a rate of 6700 samples per second for 12 seconds All tests are pcrfomied with out rotor spinning EXPERIMENTAL PROCEDURE The rotor is carefully centered within the damper clearance and the valves in the oil and air feed lines are set to the desired mixture composition The air and oil flow rates as well as the val ues of supply pressures and temperature are recorded for the com putation of the air volume fraction The system fundamental nat ural frequencies measured by impact tests under dry conditions are equal to 284 Hz and 30 Hz in the horizontal and vertical directions respectively The difference is due to asymmetry in the squirrel cage stiffness as demonstrated earlier by static load meas urements of the system exibility 5 The test system has two DOF n2 Thus two independent excitations are required to compute all four coefficients of the impedance matrix Impact loads in the horizontal X and verti 743 a u Flg 2 Conlcal mode shape of the rotor cal Y directions are sufficient to perform the identification pro cedure Eight impacts are exerted on each direction for every mix ture condition and the time traces of forces and displacements are stored The impact forces are applied at the SFD journal and the shaft displacements are measured near the end disk Equivalent X and l displacements at the SFD location are computed using the conical mode of motion with a pivot at the bushing as depicted in Fig 2 A DFT transform is applied to the dynamic displacements and loads and the resulting spectra are regrouped into eight sets each one containing the data from the X and Y impacts Equation 6 is then employed to compute the impedance elements u bf am hi for each data set at the discrete values of frequen cy Then the eight discrete functions corresponding to each impedance coef cient are averaged to render a single frequency function in which the noise not related to the load excitation is reduced Note that using the average of the impedance andt or flex ibility transfer functions instead of computing the transfer func tion from frequency averaged responses and excitations elimi nates the requirement for repetitive excitations thus allowing for the use of hand held impact hammers or the combination of dif ferent types of excitations Figure 3 shows typical time variations of the applied force and displacement responses and their corre sponding DFTs for one case of impact excitation in the X direc tion The measurements include a short prem39gger and contain the full span of the transient motions thus avoiding leaking effects on the DFT transforms Figure 3 also shows the excitation to have a wideband spectrum that covers me whole range of frequencies of interest The IVF parameter identification method Eq I l is applied to the averaged flexibilities over a selected range of frequencies around the fundamental natural frequency of the system In this case the selected range goes from 81 Hz to 488 Hz and includes the peak response resonance region The process is repeated for six different lubricant mixture compositions ranging from pure oil to 100 percent air The IVF identi cation process renders esti mates for the system force coef cients MU CU and Kutiljley as functions of the air volume content in the mixture These are equivalent system parameters referred to the location of the SFD middle plane 744 S DIAZ AND L SAN ANDRES 006 I I Z 004 J a b 002 o 4 l u l I J I d a g E E K x 2 a 4 I a I l h Q E E t gt 2 0 AL 3 20 to so Isec w HZ F 3 Typlcal Impact excitation In the X direction and response dis placements X and Y In time and frequency domains 1 000 J x O O or Flexibility umN ldentiflcation range a l l 10 20 30 40 Frequency Hz Fig 4 Measured and approximated system flexibllltles Air volume fraction 36 TEST RESULTS Figure 4 depicts with symbols the exibilities EUiJXYmeas ured for an airoil mixture volume content of 86 percent as a func tion of the excitation frequency The continuous lines represent the exibilities calculated with the estimated system parameters T he experimental values represent the averages from multiple impacts as discussed before Note that the crosscoupled exibil ities are at least one order of magnitude lower than the direct sys tem l lexibilities Correlations between the measurements and the analytical curve t functions are computed for each direct and crosscoupled exibilities to provide a measure of the goodness of the approximation All correlations range between 94 percent and 98 percent demonstrating the effectiveness of the IVF method Furthermore the coherence of the direct displacements to the exerted loads shows values near unity for the range of frequencies considered Figures 5 through 7 depict the estimated dynamic force coef cients acting at the damper location The values for an air volume fraction of one ie pure air or dry condition represent solely the effect of the support structure and rotor inertia without any influence of the squeeze lm These coefficients identi ed earli er by other means serve to validate the dynamic measurement process and identification method The direct inertia coefficients 4 N Inertia kg o 02 04 06 08 1 Air volme fradion Fig 5 Equivalent inertia coefficients vs air volume fraction StiffnesskNm Fig 6 Equivalent stiffness coefficients vs air volume fraction 200 150 0 Ca 390 100 E 4 cxy g 50 a Cm 3 o x cyy g a 50 f 0 02 04 06 08 1 Air volume fradlon Fig 7 Equivalent damping coefficients vs air volume fraction are detemiined by weighing the shaft disk and journal and using simple geometrical relations to evaluate the equivalent inertia at the SFD location The value calculated by this procedure is 402 kg and somewhat lower than the magnitudes identi ed from the dynamic response tests The direct stiffness of the elastic damper support in the horizontal direction XXX is determined by apply ing static loads with a dynamometer and recording displacements with a dial gauge indicator The measured value is KXX ISO kNm The equivalent structural damping is estimated from the logarithmic decrement 0f the dynamic response to an impact The direct damping coef cient for no lubricant is estimated as 223 Nsm Figure 5 shows the direct and crosscoupled inertia coef A Method for Identi cation of Bearing Force Coef cients and Its Application to a Squeeze Film Damper with a Bubbly Lubricant 745 cients estimated by the IVF method as a function of the mixture air volume fraction At a volume fraction of one ie pure air the VF method con rms the estimations of mass coef cients per formed by weighing the parts The results also show that no sig ni cant uid inertia is introduced by the SFD since the system direct inertia coefficients MXX Mquot remain invariant when oil ows through the damper lands The crosscoupled inertia coef cients Mxy Myx are nearly null in all test cases The estimated IVF stiffness coefficients 1K0 Fm are depict ed in Fig 6 for air volume fractions ranging from zero pure oil to one pure air The measurements for the dry condition con rm the static measurements of the structure characteristics No appreciable change is observed in any of the stiffness coef cients direct or crosscoupled when oil is fed to the damper The cross coupled stiffnesses are nearly zero though de nitely negative in all tests The vertical direct stiffness is slightly larger than the hor izontal one which agrees with the higher natural frequency meas ured in the vertical direction The average values and maximum percent variation for the stiffness and inertia force coefficients are XXX 1463kNm 43 KW 1691 kNm 41 K 223 kNm 387 K 115 kNm 242 M 47 kg 23 MYY48 kg 31 M 062 kg 434 Mquot 025 kg 524 Figure 7 depicts the variation of the system damping coef cients CiFm as the air volume content in the mixture increas es The measurements of the dry direct damping coef cients coincide with the preliminary tests based on the system logarith mic decrement Predicted values of the SFD damping coef cients for the pure oil condition centered journal and a full lm extent are equal to 17 3 h LD CXX Cy City ny 0 IOONsm 14 These values are very close to the identi ed viscous damping coef cients The estimated test crosscoupled damping coef cients are rather small most likely within the uncertainty of the measurements As expected the direct damping coef cients CXX Cquot vary signi cantly with the airoil mixture composition However contrary to expected results the direct damping coef cients increase steadily as the air volume fraction rises to a mix ture with 50 percent air content For larger concentrations of airoil volume the direct damping coef cients decrease rapidly towards their dry value The unusual damping coef cients identi ed imply an increase in the effective viscosity of the lubricant mixture for small air vol ume contents Chamniprasart et a1 18 provide a fundamental analysis and limited empirical evidence verifying this phenome non The present authors speculate that the nature of the impact tests generates too fast system transient responses which may pre vent the mixture compressibility from affecting the generation of squeeze lm pressures or the overall damping coef cients It may also be possible that since the SFD is open to ambient on both sides the air in the mixture is expelled from the film earlier than the oil thus resulting in a lubricant with a lower air content than the one measured in the supplied mixture Diaz and San Andres 8 4 15 detail measurements of damping coef cients in a SFD performing sustained circular cen tered orbital motions at various whirl frequencies In these exper iments the SFD force coefficients steadily decrease as the air con tent increases in the lubricant mixture These references reveal the complexity in the structure of bubbly flow elds and their effects on SFD force performance CONCLUSIONS The instrumental variable lter IVF method proves a reliable tool for the identi cation of bearing force coef cients The gener al formulation presented easily allows for extension of the method to account for support exibility or even shaft exibility when the equations of motion of a system need to be established experi mentally Application of the IVF renders the inertia stiffness and damping matrices of a linear system according to the selected degrees of freedom However the selection of the appropriate degrees of freedom is not always evident thus representing the most critical pant of the parameter identi cation process The excitation force employed is also an important factor Many options are available but the impact force stands out because of the ease of its implementation and its wide frequency spectrum The IVF method is applied to the identi cation of system force coef cients in a small test rotor supported on a squeeze lm damper SFD lubricated with a mixture of air in oil The meas urements show that the SFD does not introduce any signi cant amount of stiffness or inertia to the structural system The cross coupled damping coef cients are also negligible in all test cases A curious trend is unveiled for the direct damping coef cients CXX Cw Instead of a monotonic decrease for increasing air vol ume fractions the direct damping coef cients increase slightly up to a lubricant composition of about 50 percent air in volume where they reach a maximum Further increase of air content reduces the damping coef cients until they reach the dry damp ing value for a pure air condition The present results con rm that the amount of damping provided by a SFD is greatly affected by air entrainment However it is suspected that the increased vis cosity for low air volume fractions will not be enough to produce an increment of the actual damping in an operating SFD with sus tained whirl motions of signi cant amplitude and where the mix ture compressibility effect is of utmost importance ACKNOWLEDGMENTS The support of the National Science Foundation is gratefully acknowledged The rst author also acknowledges the support of CONICIT Consejo Nacional de Investigaciones Cientr cas y Tecnol gicas and Universidad Simon Bolivar Venezuela Thanks to Mr C W Karstens undergraduate student who performed most of the experimental work 746 S DIAZ AND L SAN ANDRES REFERENCES 1 Robinson M Arauz G and San Andres L A Test Rig for the Identi cation of Rotordynamic Force Coef cients of Fluid Film Bearing Elements ASME Paper 95GT431 1995 2 Ransom D L Identi cation of Dynamic Force Coef cients of a Labyrinth and Gus Dumper Seal Using Impact Load Excitationsquot Master Thesis Texas AampM University December I997 3 Fritzen C Identi cation of Mass Damping and Stiffness Matrices of Mechanical Systems ASME Paper 85DET9l I985 4 Massmann H and Nordmann R quotSome New Results Concerning the Dynamic Behavior of Annular Turbulent Sealsquot NASA CP 2409 in Prac of the Instability in Rotating Machinery Workrhop Carlson City pp 179194 I985 5 MullerKarger C M and Granados A L Derivation of Hydrodynamic Bearing Coef cients Using the Minimum Square Methodquot ASME Jaun of Trib H9 4 pp 802807 I997 6 Diuz S E and San Andres L A Measurements of Pressure in a Squeeze Film Damper with an AirOil Bubbly Mixturequot Trib Trans 411 2 pp 282288 1998 7 Rouvas C Parameter Identi cation of the Rotordynamic Coef cients of High ReynoldsNumber Hydrostatic Bearingsquot PhD Dissentation Texas AampM University College Station TX I993 8 Diaz S E Experimental Parameter Identi cation of a nxn Linear System Internal Research Progress Report Rotordynamics Laboratory Texas AampM University College Station TX I997 9 Ransom 0 Li 1 San Andres L and Vance J M Experimental Force Coef cients for a TwoBladed Labyrinth Seal and a FourPocket Damper Seal ASME Paper 98THb28 I998 10 Walton 1 Walowit 1 Zorzi E and Schrand J Experimental Observation of Cavitating Squeeze Film Dampersquot ASME Joun of Trib 109 pp 290295 I987 11 Zeidan F Y and Vance J M Cavitation Leading to a Two Phase Fluid in a Squeeze Film Damperquot Trib Trans 32 I pp 100104 1989 12 Zeidan F Y Vance J M and San Andres L A Design and Application of Squeeze Film Dampers in Rotating Machineryquot in Proc of the 25th Turbamachinery Symp Texas AampM University College Station TX pp 169 188 1996 13 Childs D Turbamachinery Rotardynamics John Wiley amp Sons NY I993 14 Diaz S and San Andres L Reduction of the Dynamic Load Capacity in a Squeeze Film Damper Operating with a Bubbly Lubricantquot ASME Paper 98 GT109 I998 15 Diaz S and San Andres L Effects of Bubbly Flow on the Dynamic Pressure Fields of a Test Squeeze Film Damper ASME Paper F EDSM985070 in Proc of the 1998 ASME Fluids Engineering Division Summer Meeting Washington DC June I998 16 Karstens C W Effects of Air Entrainment on the Damping Coef cients of a Squeeze Film Damperquot Senior Honor Thesis Texas AampM University I997 17 San Andres L and Vance 1 Effect of Fluid Inertia on Squeeze Film Damper Forces for Small Amplitude Circular Centered Motionsquot ASLE Trans 30 pp 6976 1987 18 Chamniprasart K AlSharif A Rajagopal K R and SZeri A Z Lubrication With Binary Mixtures Bubbly Oil ASME Jaun of Trib 5 pp 253 260 I993

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