MODAL ANALYSIS AND TESTING OF ROTATING MACHINES FOR PREDICTIVE MAINTENANCE: EFFECT OF GYROSCOPIC FORCES

Transcription

1 MODAL ANALYSIS AND TESTING OF ROTATING MACHINES FOR PREDICTIVE MAINTENANCE: EFFECT OF GYROSCOPIC FORCES A. Vansteenkste; M. Loccufer; B. Vervsch; P. De Baets; Ghent Unversty, Belgum Abstract Through the year s preventve mantenance, whch s based on vbraton measurements has grown n mportance to reduce costs. However n ndustry nowadays there s stll a great need for good analytcal predcton models that descrbe the dynamcs of rotatng structures. In ths paper t s frst shown how to buld a model for for undamped gyroscopc systems. Secondly the model s analysed and used to make a parametrc study for the desgn a test rg. It s found that the thrd and the fourth egenfrequency changen functon of the rotaton speed due to the gyroscopc effect. Fnally the expermental modal testng of rotatng structures s dscussed. The methods to excte a rotatng structure n order to obtan the modal parameters are studed. Keywords: Rotor dynamcs; expermental modal testng; Campbell dagram; gyroscopc effect; 1 INTRODUCTION Rotatng machnes are wdely used n our modern lfe, rangng from washng machnes to steam turbnes and aeroplanes. Through the years there has been a tendency to faster and better performng rotatng machnes wth lower vbraton levels. However, to fulfll these desgn specfcatons, t s necessary to model, analyse and mprove the dynamcs of rotatng structures, because the effcency s mostly nfluenced by the small gaps between the rotatng en the statc parts. Another reason why dynamc models are needed s for predctve mantenance, whch s based on vbraton measurements. So, n order to evaluate those measurements t s mportant to have good dynamc models. However n ndustry there s stll a great need for accurate analytcal predcton models that descrbe the dynamcs of rotatng structures, such as the relablty, stablty and response levels. A problem s to obtan the parameters of the dynamcal model expermentally. For non-rotatng structures t s well known how to obtan the modal parameters expermentally. For rotatng structures ths s not the case [1]. Another problem s that t s not easy to model rotatng machnes analytcally, as they are mostly too complex to model n a relable way. Therefore, t s necessary to desgn test rgs havng an easer constructon and easy operatng and boundary condtons. Ths paper shows how to model an undamped gyroscopc system and whch boundary condtons and assumptons have to be made. Furthermore, a model for undamped gyroscopc systems has been bult n Matlab. Smulatons are done to obtan a set of parameters to desgn a test rg. Fnally, the last part s of ths paper s devoted to expermental modal testng of the test rg. 2 MODELLING ROTATING SYSTEMS [2] 2.1 Assumptons for rotatng systems A frst assumpton s that the vbraton levels are suffcently small such that lnear models can be used. Secondly the number of degrees of freedom has to be as low as possble to lmt the computaton effort., Fnally ts assumed that the rotatng structure contans only sotropc rotatng elements and that there s a quas sotropc bearng support. Consequently the modes are planar. 2.2 General Model for rotatng systems The equatons of moton for a rotatng system, wth the ncluson of the gyroscopc and the centrfugal forces have the followng structure: M q& + ( G + C) q& + ( K H ) q = f ( t) (1) wth respectvely mass matrx M, gyroscopc matrx G, dampng matrx C, stffness matrx K and the matrx of the centrfugal forces H.

2 Due to the gyroscopc effect, the egenfrequences are not all constant anymore, but some are dependng on the rotatng speed. Ths can be shown on a Campbell dagram or a whrl speed map, as for example fgure 1. It can be seen that an egenfrequency s ncreasng for forward whrl (FW) and decreasng for a backward whrl (BW). 2.3 Modellng undamped gyroscopc systems Fgure 1 Campbell dagram To study thethe nfluence of the gyroscopc effect on the vbraton modes and to obtan a better nsght nto the gyroscopc effect and the mathematcal complexty s lowered by neglectng the effect of the centrfugal force and the dampng. Due to these smplfcatons the equaton of moton s reduced to equaton (2). 2.4 Fnte element model of the system M q& &+ Gq& + Kq = f (t) (2) To descrbe the gyroscopc system, the shaft has been dvded nto a number of elements. The number of mass statons equals thenumber of elements + 1 and s based on followng concerns. The choce of the number of elements s a compromse between enough freedom to choose exctaton and measurement locatons and lmtng the number of modelled modes. Fgure 2 shows the chosen 10-element model. z y x Fgure 2 - fnte element model wth 10 elements (rght-handed coordnate system) 2.5 Obtanng matrces for a shaft fnte element A mass staton contans two lnear dsplacements: x and y en two angular dsplacements θ x θ y resultng n four DOFs: [x y θ x θ y ]. Hence one element (contanng two mass statons) contans 8 DOF: [x 1 y 1 θx 1 θy 1 x 2 y 2 θx 2 θy 2 ] whch can be seen n fgure 3.

3 Fgure 3 - Basc rotor element: angular and lnear dsplacement n x- and y-drecton [2] By usng the consstent mass method, the dstrbuton of the acceleraton of the ponts between the nodes of the elements s gven by a shape functon that has the same form as a statc deflecton. By usng the Lagrange approach the consstent mass matrx for one element can be obtaned. Frst the translatonal shaft knetc energy of the th element s calculated as equaton (3) shows.. T = 1. 2 M L L ( x² + y& ²) dz 0 & (3) Substtutng ths nto the acceleraton term of the Lagrange equaton (4) wll yeld the mass matrx. d dt T ( ) = q q& [ M] {& } Where q denotes the vector of generalzed coordnates., = 1, 2 8 (4) The polar moment of nerta s lumped n two equal parts at the elements end ponts to obtan the skewsymmetrc gyroscopc matrx. The consstent stffness matrx for one element can be smlarly obtaned by usng the potental energy. In order to derve the [M], [G] and [K] matrces for the complete free rotor, the element matrces have to be assembled. Other non-structural mass and nerta, such as that of the rotatng dsc can be added by lumpng at mass statons. To obtan the complete [K] matrx of the fxed rotor, the bearng stffness has to be added n the mass statons as well. 2.6 Transformaton nto state-space form [3] In order to solve the generalzed egenvalue problem, the model s transformed nto a state space form. q( t) x( t) = q& ( t) The equaton of moton s now changed nto a frst order dfferental equaton (5) or (6). K 0 K 0 x& x M + K G = f ( t) Mˆ Gˆ M ˆx + Gx ˆ = Or & ) (6) In case of a model wth 10 element, ths results n a model wth 2.4. (10+1) = 88 DOF. fˆ( t fˆ ( t) (5) 3 ANALYSIS AND PARAMETRIC STUDY In the fgure 4 the dfferent mode shapes. The two frst egenfrequences are equal and ther correspondng egenvectors represent each a dsplacement n x- or y-drecton, that descrbe a sphercal cylnder moton. The thrd and fourth egenfrequences are not equal anymore and ther egenvectors represent each a

4 moton n x- and y-drecton, that descrbe a concal cylnder moton. In x and θ y drecton the 2 nd en 4 th mode are predomnant. However n y and θ x drecton the 1 st en 3 rd mode are predomnant. Fgure 4 - Mode shapes n the nodes To llustrate the motons, the dsplacement of the frst node s shown n functon of tme n fgure 5. Fgure 5 Moton of the frst node n functon of tme To desgn the test rg some parameters have to be determned by smulaton n Matlab. The shaft s chosen at a length of one meter. The parameters that are nvestgated are the shaft dameter, the mass and the dmensons of the dsk, the locaton of the dsk, the stffness of the bearngs and of course the nfluence of the rotatng speed. The fxng method of the motor s also analysed.

5 3.1 Rotatng speed When the rotaton speed s ncreased, a decrease of the thrd egenfrequency and an ncrease of the fourth egenfrequency s observed. It can be concluded that the thrd and the fourth mode are dependng on the rotaton speed due to the gyroscopc effect. The frst two egenfrequences are ndependent of the rotaton speed. It can be concluded that t s preferred to have a speed range of the motor that s s suffcently hgh order to see the gyroscopc effect through the dfference between the thrd and the fourth egenfrequency. 3.2 Shaft thckness The thckness of the shaft s preferred to be as small as possble to obtan hgher egenfrequences, whch ncrease the gyroscopc effect. However the shaft needs a certan thckness n order to resst the bendng moment due to the mass of the dsk. 3.3 Dsk Increasng the mass moment of nerta of the dsc ncreases the gyroscopc effect, whch can be seen n a decrease of the 3 rd and an ncrease of the 4 th egenfrequency. Increasng the mass of the dsc lowers the egenfrequences and demands a stffer shaft n order to reduce the stresses due to bendng. Addng more dsks s found to reduce the gyroscopc effect. It can be concluded that n order to ncrease the gyroscopc effect, the dsk needs a hgh mass moment of nerta and a low mass. 3.4 Stffness of the bearngs The rato of the stffness of the bearngs to the stffness of the shaft determnes the mode shape. 3.5 Fxaton of the motor When the motor s fxed coupled to the shaft, ths can be modelled by puttng a second dsk n an added node, whch s not needed. To elmnate ths, the motor s coupled to the shaft by a flexble couplng. The test rg wll be powered by an electrcal motor that wll be chosen n functon of the mass moment of nerta of the dsk. 4 EXPERIMENTALLY MODAL TESTING [1] Expermental modal testng can be dvded nto two steps. Frstly, there s an acquston of response data, whch s typcally done by measurng the frequency response functons (FRFs). Secondly, the modal parameters are found by curve fttng of the measured response functons. 4.1 Acquston of the response data To acqure response data the test rg has to be excted. Ths s far more dffcult than the exctaton of nonrotatng structures due to the rotaton. A frst method s the exctaton by a controlled unbalance. It s a smple method, because no specal mechancal mechansms are needed. A drawback of ths method s the dffculty of mantanng a constant frequency. A second method s the exctaton wth an mpulse hammer. Ths s also an easy method because of the lack addtonal mechancal components. Another advantage s that several modes are excted smultaneously. However t s not easy to repeat wth the same proper force and drecton for several tmes. Another dsadvantage s a lot of the nose on the sgnal (a hgh SNR). A thrd method whch reduces the drawbacks of the prevous method s the exctaton wth an electrodynamc shaker that s connected by an addtonal bearng. Wth ths method the force and frequency can be controlled, whch permts to repeat the measurements. There s also less nose on the sgnal, whch results n an mproved qualty of the sgnal. A drawback of ths method s the addtonal bearng that has to be attached to the shaft and that s affected by the moton of the shaft. Thereby the force s also functon of the response, whch can be modelled wth feedback and results n a change of the FRF. 4.2 Obtanng the modal parameters The modal parameters are obtaned by curve fttng, whch s done n the frequency doman. The FRF has the form gven n equaton (7): N Ar Ar E2 H ( ω) = + + E0 + ω λ ω λ r ² r= 1 r ω (7)

6 Where E 0 and E 2 are resdual terms n order to compensate the modes that are out of range. For nonrotatng structures the resdue matrx A r s symmetrc [3], however for rotatng structures A r s not symmetrc anymore. 4.3 Smulaton n Matlab Wth the model made n Matlab, exctatons are smulated on dfferent nodes and hence the correspondng mpulse responses and FRF s (frequency response functons ) are obtaned. Frst t can be concluded that when there s measured n the same drecton as the exctaton drecton, there s a peak n the FRF due to the frst or the second mode dependng on the drecton. Secondly t can be concluded that when there s measured n the other drecton than the exctaton drecton, the thrd and the fourth mode peak, whch s mportant to measure the gyroscopc effect. The hghest relatve peaks of the 3 rd and 4 th mode are observed when excted n the 3 rd node and measured n the 5 th node or vce versa. In fgure 6 the FRF s shown for an mpulse n x-drecton appled to node 3 and measured n y-drecton n node 5. It can be seen that the 3 rd mode (around 84Hz) and 4 th mode (around 143Hz) are excted. Fgure 6: FRF for an mpulse n node 3 ( n x-drecton), measured n node 5 ( n y-drecton). 5 CONCLUSION In ths paper, a model for undamped gyroscopc systems has been bult n Matlab. Smulatons have been made n order to fully understand the mechansm and to become a set of parameters to desgn a test rg. Wth those parameters a test rg wll be desgned n order to compare the measured modal parameters wth the parameters calculated wth the model n Matlab. Wth those results the expermental modal analyss methods can be evaluated and may be mproved. A frst challenge wll be to fnd the best exctaton method. Three exctaton technques have been found n the lterature and ther potental and practcal lmtatons and have been dscussed. Secondly the vbratons have to be measured usng accelerometers. Another dffculty wll be the sgnal processng and the correct nterpretaton of the measured data. In general t can be concluded that ths topc s not yet fully known and that there are stll some challenges that have to be solved before t wll be possble to apply expermental modal analyss to any type of rotatng machnes.

7 6 NOMENCLATURE M = mass matrx C = dampng matrx G = gyroscopc matrx K = stffness matrx H = centrfugal force matrx f(t) = force matrx 7 REFERENCES [1] Bucher I. and Ewns D.J. Modal analyss and testng of rotatng structures. The royal socety 2001; 359, [2] Adams Maurce L. Rotatng machnery vbraton, from analyss to troubleshootng. Marcel Dekker 2000; [3] Loccufer M. Cursus Mechansche trllngen. Unverstet Gent [4] Brte. Development of valdated structural dynamc modellng and testng technques for vbraton predctons n rotatng machnery. 1996