Effect of insulation paper ageing on the vibration characteristics of a winding disc

Transcription

1 Pper Number 45 Proceedings of COUSTICS 11-4 November 11 Gold Cost ustrli Effect of insultion pper geing on the vibrtion chrcteristics of winding disc Ming Jin nd Jie Pn School of Mechnicl Engineering University of Western ustrli W ustrli BSTRCT Vibrtion nlysis hs been pplied to power trnsformer condition monitoring in the lst few decdes. It uses the chrcteristic vibrtion of the trnsformer to detect nd dignose trnsformer filures. Becuse the winding insultion pper degrdtion is very common nd serious power trnsformer filure this pper is concerned with the effect of geing of winding insultion pper on the vibrtion of simplified winding model comprising single lyer of concentric circulr rings wrpped with insultion pper. The insultion pper undertkes high therml nd electricl pressure in n opertionl trnsformer nd the verge moleculr of cellulose chins decreses with ge which cuses both the insultion bilities nd mechnicl strength of the pper to degrde. By testing the vibrtion of the concentric rings with different geing ppers it is found tht the nturl frequencies of the rings shift to low frequency end when the mechnicl strength of the pper decreses. mechnicl model of the rings with pper is provided here to identify the reltionship between the pper mechnicl properties with their corresponding vibrtions. Bsed on the work reported in this pper it is believed tht the geing extent of winding insultion pper cn be detected by monitoring the vibrtion of the winding. 1. INTRODUCTION The power trnsformer is one of the most importnt components in the power industry. Filures of serving power trnsformer my cuse huge economic loss directly nd indirectly. lthough the primry power trnsformer filure is the lod tp chnger problem (4%) nd the power trnsformers disc-type winding problem is secondry (15%- %) the ltter filure is much more lethl [1 ]. Usully serious power winding filure is unrepirble nd the only solution is replcing the filed trnsformer which costs millions of dollrs. Therefore monitoring the performnce of power trnsformer disc-type windings nd preventing them from filure is prcticlly significnt. One pproch in condition monitoring of power trnsformers involves detecting trnsformer winding filures by nlysing the chrcteristic vibrtion of the winding structure used in lst few decdes. The distinct merit of this vibrtion method llows continuous noninvsive on-line monitoring nd identifiction of the trnsformer problems instntneously. These dvntges ttrcted mny reserchers working on this method. Grcí nd his group [] indicted the generl reltionship of winding vibrtion with its input voltge nd current fter series of vibrtion tests on dry power trnsformer winding without cooling oil. They lso provided some convincing evidence showing tht the vibrtion mesured from the trnsformer tnk cn be used to detect some winding filure modes such s winding deformtion. Similr vibrtion experimentl results nd conclusions cn be found in other ppers [4 5]. However considering the prcticl winding filures it is reported tht bout hlf of filures re cused by the insultion problems [1]. Despite this it ppers tht no significnt reserch discussing this winding insultion issue using winding vibrtion hs been conducted. Therefore this pper focuses on fundmentl reserch of the effect of winding insultion pper geing on winding vibrtion. Considering the complicted structure of disc-type power trnsformer winding this pper ims to understnd the effect of insultion pper geing on the winding s vibrtion using reltively simple winding model comprising single lyer of concentric circulr rings. This is becuse the disc-type power trnsformer winding is formed from series of discs nd ech disc cn be treted s one lyer of concentric circulr rings. Becuse the min exciting electromgnetic force pplied on the winding is induced by the current in the winding most of winding vibrtion energy is concentrted in the rdil direction. Therefore this pper only discusses the in-plne vibrtion of the rings. In the first prt of this pper the mechnicl properties of the geing insultion pper will be mentioned s they re considered to be key influencing prmeters in the winding vibrtion. Then the effect of insultion pper geing on winding vibrtion is simulted experimentlly by two single lyer winding discs with ged pper. Finlly mechnicl model of concentric circulr rings is built to explin the experimentl phenomen theoreticlly.. GEING OF WINDING INSULTION PPER The power trnsformer winding comprises of discs formed by severl lyers of insultion pper wrpped tightly round the winding wire but not solenoid formed. This design is helpful for the winding to undertke lrge current nd provide enough insultion bility. However winding insultion pper in the serving power trnsformers is ged over time cused by the internl high electricl nd therml pressures in the trnsformer tnk. The insultion bility of the pper degrdes over time nd winding with filed insultion pper could be thoroughly dmged by short circuit ccidents. For this reson the insultion properties of winding insultion pper hve been the topic of concern by electricl scholrs for very long time. To define the insultion bility of the pper the Degree of Polymeristion (DP) is used. It is the verge number of glucose units per cellulose molecule. s the pper ges the polymer chins of the cellulose molecule is crcked so tht the DP vlue decreses. s result glucose is degrded to some smll molecule products such s furfurldehyde or furn. The DP vlue for new insultion pper is greter thn 8. It is believed tht when this vlue is lower thn the pper could be defined s filed which mens its insultion bility is insufficient [6 7]. Most of the previous reserch on insultion pper geing focuses on the electricl properties of the pper with DP chnges. This pper however is more concerned with the mechnicl properties of the geing pper becuse it my ffect the vibrtion behvior of the winding. coustics 11 1

2 -4 November 11 Gold Cost ustrli Proceedings of COUSTICS 11 n ccelerting geing experiment is designed for obtining the ged pper smples. Severl new insulted winding copper wire strips (1cm) re immersed in the trnsformer s cooling oil in seled glss bottles. ll the bottles re heted to 16ºC in furnce for between 7 nd 4. ccording to Hill s pper [7] the pper t such high temperture should be thoroughly filed (DP<) in 4. fter geing the strips were tken out of the oil nd dried t 6ºC for 6 hours. Then the pper smples were removed from the wire for the mechnicl tests. Force (N) new 1 geing 4 geing Compression (mm) Figure : Out-of-plne force/compression curves of pper New pper ged pper (4 ) Figure 1. Cellulose of insultion pper smples before nd fter the geing experiment (smple size pprox. 1µm 1 µm ) s mentioned before when the pper ges some of the polymer chins of cellulose molecules brek. Physiclly the brekge cn be directly observed by using microscope. Figure 1 compres the cellulose of the pper before nd fter the geing experiment. lthough the two pictures come from different pper smples the phenomen of cellulose re observble lmost everywhere in the smples. In the new smple most of cellulose molecules re unbroken. But some obvious brekges (highlight by the red circle) could be found in the ged pper. s result it is logicl to infer tht the mechnicl strength of the pper must chnge when it ges. Tensile Force (N) new Extension (mm) Figure : In-plne tensile force of pper Figure shows the in-plne ultimte tensile force of single lyer of pper (re of cross-section pprox. mm ) before it crcks. The pper is clmped t both ends nd drgged in opposite directions until it is torn. One cler trend is the ultimte force decreses with the ge time. However nother interesting phenomenon is the stiffness of the pper in the inplne direction does not decreses but increses. possible explntion for this result is tht becuse the number of polymer chins of cellulose molecule decrese fter geing the ultimte tensile force the pper cn undertke lso decrese. However for ech cellulose molecule it hrdens up cused by the high therml pressure during the geing test nd the cooling processing fter the geing. Similr to cold rolled steel it becomes stiffer but more brittle. s result the in-plne stiffness of the pper increses before the ultimte tensile force point. It lso provides nother reson why the pper is esy to crck fter geing. On the other hnd Figure gives the force/compression curves for 1 lyers of pper (re of crosssection is 4 mm ) in the out-of-plne direction. For clrity only the results from the new 1 ged nd 4 ged pper re given. From these curves it is found tht the out-ofplne stiffness of the pper (see Tble 1) decreses with the geing time. This phenomenon could be explined by Pick Up Sticks gme: piece of insultion pper consisted of thousnds of cellulose molecules cn be seen s lots of sticks combined together. For new pper every stick is reltively long nd supported by ech other. Therefore the pper is hrd to compress nd ppers stiff. When the pper ges s the cellulose molecules re crcked ech stick breks into severl short sticks the support effect wekens nd it is esier to compress. When the stiffness chnge hs been explined logicl supposition is the dmping effect of the ged pper my reduce cused by the sme geing phenomenon. The dmping of the pper minly comes from viscous effect between pper cellulose molecules. Becuse the verge contct surfce of crcked cellulose molecules is smller thn the perfect cellulose molecules the viscous effect between them decreses. This mens the dmping coefficient of the pper should reduce when the pper ges.. WINDING DISC WITH GED INSULTION PPER From the results of the previous section it is cler tht the mechnicl properties of the insultion pper vry with ge. s the pper is one of the min components of the winding it is supposed tht the chnge in mechnicl properties of the insultion pper should ffect the winding s mechnicl vibrtion. To investigte this nother ccelerting geing experiment is implemented using winding disc wrpped with insultion pper. s one element of disc-type winding the results of the winding disc could represent the vibrtion chrcteristic of the winding especilly in the rdil direction. coustics 11

3 Proceedings of COUSTICS 11 geing time In-plne ultimte force (N) In-plne stiffness (1 5 N/m) Out-ofplne verge stiffness (1 7 N/m) Tble 1. geing properties of the insultion pper New Figure 4. Two winding discs with insultion pper (fter the ccelerting geing experiment) -4 November 11 Gold Cost ustrli tht too mny holes my ffect the mechnicl properties of the insultion pper the disc vibrtion is only mesured t limited loctions. Figure 5 represents the vibrtion mesurement setting. The ngle between θ nd θ is bout 5 degrees. For disc 1 the vibrtion is mesurement t loctions θ 1 to θ 8. For disc the vibrtion is only recorded t loctions θ 1 to θ 4. Figure 6 shows the mesured rdil vibrtion of ech disc t different ge times. For clrity 1dB off-set is pplied one by one except the top curve in ech digrm. The sme experimentl fetures could be observed from different discs or different mesurement loctions. The min fetures of the mesured vibrtion re: 1. The nturl frequencies of the disc shift to the low frequency end during the geing process. The detils of this shift re given in Tble nd the top digrm of Figure 7. Besides the sme shift trend the sme nturl frequency of these two discs shift by similr extent in percentge nd the movement of the lower order nturl frequency is bigger thn the higher order one.. The frequency bnd of the disc resonnce peks becomes nrrower when the geing time increses. From Tble nd the bottom digrm of Figure 7 it is found tht the clculted system dmping rtios bsed on db decy method decrese with the geing time especilly the first two modes from 7 to 1. To improve the ccurcy of the experiment two different copper winding discs (Figure 4) re used. The prmeters of these discs re listed in Tble. Similr to the pper geing experiment these two discs re immersed in trnsformer cooling oil in seled continer. The geing experiment is performed t 16ºC for totl of 8. During the experiment the vibrtions of the discs re mesured every 7. Before ech vibrtion mesurement the discs re removed from the oil nd dried t 6ºC for 6 hours nd then they cool to room temperture. Tble. Prmeters of tested discs Disc 1 Disc Number of turns 5 6 Dimeter of outside turn (mm) Dimeter of inside turn (mm) Thickness of tpe (mm) Figure 5. Schemtics of vibrtion mesurement setting Width of tpe (mm) t the geing temperture 16ºC the physicl properties of Lyers of pper for ech turn 4 4 copper wire should be stble. ll the vibrtion mesurements re implemented t room temperture so tht the geometricl chnge due to therml expnsion nd construction of the copper During the vibrtion experiment ech disc is hung on metl disc could be negligible. Therefore it is resonble to infer tht stnd nd excited by hmmer impulse force pplied t the bove experimentl phenomen re cused by the geing of loction θ on the outside turn opposite the hnging point. Then insultion pper. In Section it is known tht: the stiffness nd the rdil vibrtion of the disc is collected from severl the dmping coefficient of insultion pper in the out-of-plne loctions θ 1 to θ N on both the outside nd inside turns by direction decreses with the ge. Therefore the system stiffness ccelerometers. It is worthwhile to note tht for mesuring the nd dmping rtio ξ of the disc with insultion pper lso vibrtion of the disc the ccelerometers must be ttched on decreses cused by the pper degrdtion over time. In the surfce of the copper tpe not on the insultion pper. mechnicl sense we know tht the system nturl frequencies From experience of former experiment the disc vibrtion f should reduce when system stiffness decreses (feture 1). recorded from the pper is not relible t ll. Therefore severl Similrly when the system dmping rtio becomes smller s holes re cut on the insultion pper so tht the ccelerometers the pper geing the frequency bnd of these resonnce peks cn be fixed on the surfce of copper wire directly. Considering nrrows down (feture ). However s these explntions re coustics 11

4 -4 November 11 Gold Cost ustrli Proceedings of COUSTICS 11 bsed on the generl mechnicl knowledge more powerful theoreticl evidence should be provided to support the conclusion. Figure 6. Rdil vibrtions of winding discs with different ged insultion pper (top digrm: disc1 ccelerometer loction θ ; middle digrm: disc 1 ccelerometer loction θ ;bottom digrm: disc ccelerometer loction θ. 1dB off-set is pplied to ech curve except the top curve for clrity) Tble. Nturl frequencies nd dmping rtios of the discs geing time New cc. Res(dB) cc. Res(dB) cc. Res(dB) new disc 7 geing 1 geing 8 geing Frquency (Hz) Nturl frequency 1 Frequency (Hz) Nturl frequency Nturl frequency f (Hz) ξ* f (Hz) ξ* f (Hz) ξ* Disc Disc Disc Disc Disc Disc Disc Disc Nturl Frequency Shift (%) Dmping Rtio Chnge (%) -5 1st NF of Disc1 1st NF of Disc -1 nd NF of Disc1 nd NF of Disc rd NF of Disc1 rd NF of Disc Figure 7. Nturl frequency shift (top) nd dmping rtio chnge (bottom) of the discs with different geing times 4. MODELLING OF RINGS In order to give some theoreticl explntions for the disc experiment preliminrily winding disc model is built nd discussed in this section. s mention before single lyer winding disc could be pproximtely modelled s series of concentric circulr rings. For clrity the model of single ring is given first in this section. Then this single ring model cn be extended to concentric circulr rings. 4.1 SINGLE RING MODEL The vibrtion of ring hs been reserched since the 19s. Love gve the first single ring model in his book Tretise on the Mthemticl Theory of Elsticity [8]. But his model is limited to circulr cross-section nd neglects sher deflections. Since then lots of reserchers hve strived to improve this model. Regrding to the prcticl disc used in the experiment focus is directed to the in-plne vibrtion of thin ring with rectngle cross-section in this pper. Bsed on the elstic theory the in-plne vibrtion of single ring could be described by following five equtions [9]: U t V t geing time () P Q Q P M U P kz V t (1) () () *Bsed on db decy method the dmping rtio cn be clculted by f = where Δf is the bndwidth of db decy f r is the resonnce f r frequency. E V Q U Ek V U M z (4) (5) 4 coustics 11

5 Proceedings of COUSTICS 11 where (U V) nd (P Q) re the displcements nd resultnt forces in the rdil nd tngent direction respectively; M nd k z re torque nd re rdii of gyrtion of the cross-section in the xil direction; E ρ nd re the rdius Young s modulus density nd re of cross-section of the ring respectively. Bsed on the Modl Expnsion Method the vibrtion of structure could be treted s the superposition of its infinite modes. For the uniform ring its in-plne mode shpe functions cn be written s [1]: -4 November 11 Gold Cost ustrli Substituting Equtions (6) nd (7) into Eqution (1) nd using Eqution (1) nd orthogonl reltionships between the mode shpes we obtin the model coefficient C 1m F m cos( m ) ( WW m W ) 1 which together with Eqution (1) yields the system response. (1) j t U C sin( m ) e (6) 1m m1 6 j t V C cos( m ) e. (7) m m1 Substituting Eqution (4) into Eqution () nd using the mode shpe functions (6) nd (7) yields cc. Res(dB) 4 E U Em x P V V Similrly extrcting P gin by substituting Equtions (5) (6) nd (7) into Eqution () gives Em kz k z U P ( V). When the externl force F F ( ) e jt t locttion θ is not zero by combining Equtions (8) nd (9) we obtin the inplne vibrtion eqution of the ring s U WV W F 1 where W E m Em k k 4 1 z z W E Em k k z z.. (8) (9) (1) However there re two unknown vribles C 1m nd C m in Eqution (1). To solve the Eqution (1) differentiting Eqution () with respect to θ eliminting P between the resulting eqution nd Eqution (1) nd using Eqution (4) to eliminte Q gives: V V U E 1 ( U ). t (11) - model solution experimentl result Frquency Frequency (Hz) (Hz) Figure 8. Rdil vibrtion of single ring t loction θ To confirm the relibility of the model comprison experiment of copper ring is implemented. The dimeter of the ring is mm nd the remining prmeters of the ring cn be found in Tble. The experiment setting is the sme with the disc experiment. From Figure 8 it is obvious tht the model solution (in-plne vibrtion response of the ring) grees highly with the experimentl result. The smll difference my be cused by the error in the vlue for the copper Young s modulus nd the geometric structure of the ring being imperfect. 4. CONCENTRIC RINGS MODEL WITH INSULTION PPER The next step is to extend the single ring model to build preliminry model of concentric circulr rings with insultion pper to reconstruct to the phenomen in the disc experiment nd explin them. To describe the role of insultion pper in the model it is ssumed tht every two djcent rings re coupled by the insultion pper between them. Becuse the pper distribution is uniform between rings it cn be divided into N finite elements t corresponding loctions θ 1 to θ N nd ech element could be modelled s spring-dmper system (see Figure 9). For clrity only the simplest model three rings with one element is demonstrted here. The result cn be esily extended to Z rings with N pper elements where Z nd N re rbitrry integers. Expnding Eqution (11) by using the mode shpe functions (6) nd (7) gives W C (1) m C m 1 m E where m W E m 1 1. coustics 11 5

6 -4 November 11 Gold Cost ustrli Proceedings of COUSTICS 11 In the prcticl model clibrtion only finite number of M modes re used. Therefore Equtions (17) to (19) could be expnded to M independent equtions with M vribles C 11 to C C1M which re solvble. 8 6 cc. Res(dB) 4 - Figure 9. Theoreticl model of concentric circulr rings with insultion pper Still ssuming the externl force is pplied on the outside ring (Ring ) t loction θ nd the element is t loction θ 1 the vibrtion equtions of these three rings re: U W V W F 1 U W V W F B B B1 B B B U W V W F C C C1 C C C where F F k jc [ U U ] p p 1 B 1 1 FB k p jc p U 1 U B 1 1 k p jc p [ U B 1 UC 1 ] F k j C [ U U ]. C p p B C (14) (15) (16) ll the symbols represent the sme prmeters of the single ring model with subscript tht indictes the different rings. k p nd C p represent the unit out-of-plne stiffness nd dmping coefficient of ech pper element. fter similr process to the single ring model Equtions (14) to (16) cn be rewritten s W1W k jc mw C m 1 m p p cos m [ sin m ( C C ) F co ( m ) 1 1 1m B1m ] s m1 (17) cc. Res(dB) model solution experimentl result Frquency Frequency (Hz) (Hz) Figure 1. The comprison of the concentric rings model solutions with experimentl results Figure 1 compres the solutions of this concentric rings model with the experimentl results. The model solutions which re obtined from 6 rings nd 1 pper elements model simulte the disc vibrtion t the outside loction θ in the experiment (see Figure 5). The top digrm indictes this model is ble to describe the vibrtion of the single ring ccurtely when ssuming the k p nd C p equl to zero which mens the outside ring loses ll its coupling effect nd vibrtes s single ring. The bottom digrm tries to reconstruct the vibrtions of disc with ged pper. lthough there re some obvious mplitude differences the model solution provides close vlues of these nturl frequencies with the experimentl result. To further check the correctness of this concentric rings model Figure 11 compres the first two mode shpes of the disc. The blue circle is the clculted shpe from the model nd the red points re the mesured result of the experiment. The coordinte points º 9º 18º nd 7º re corresponding to the mesurement points θ θ 7 θ 1 nd θ 5 respectively in the experiment (see Figure 5). The first mode shpe is in gret greement. For the second mode lthough some differences exist between the model nd experiment the overll mode shpe is lso quite similr. WB1WB k jc mw B C Bm 1 m B p p cos m [ sin m ( C C C )] 1 1 1m B1m C1m m1 (18) WC1WC k jc mw C C C1m m 1 1 B1m C1m m1 C p p cos m [ sin m ( C C )]. (19) 6 coustics 11

7 Proceedings of COUSTICS 11-4 November 11 Gold Cost ustrli First mode shpe Second mode shpe describe its effect in the tngent direction of the disc vibrtion. The in-plne mechnicl properties of the pper my include the in-plne stiffness of the pper nd the friction between pper lyers. The friction introduced by the pper cn be prt of or even the min prt of system dmping in the disc. Therefore when the pper effect in tngentil direction of the disc is included the ccurcy of the model my be highly improved Figure 11. The normlised mode shpes of the first two modes of the disc 1 ccording to bove comprison it is resonble to sy tht this preliminry concentric circulr rings model is ble to describe some mechnicl chrcteristics of the winding disc. Figure 1 plots the model solutions using different pper prmeters k p nd C p to simulte the disc with new nd geing pper. For clrity the figure focuses on the third nturl frequency. The blue curve represents the new disc with pper prmeters k p = N/m nd C p =1 Ns/m. The red curve represents the ged disc whose pper prmeters re ten times smller thn the blue curve. The nturl frequency of the disc shifts to the low frequency end when the mechnicl properties of the insultion pper decrese representing the geing of the winding insultion pper. This trend ws lso observed experimentlly. cc. Res (db) X: 86 Y: X: 869 Y: model solution experimentl result equilibrium position new disc ged disc Frequency (Hz) Figure 1. Theoreticl simultion of winding disc with new nd ged insultion pper Two problems of this concentric rings model re worth discussing here. On one hnd the model cnnot explin the dmping effect in the experiment. Bsed on the model it seems tht the dmping chnge of the pper hrdly ffects the disc vibrtion. s result the model solution hs much nrrower resonnce peks thn the experimentl result nd the frequency bnd of these peks does not chnge with different pper dmping vlues. On the other hnd lthough the model nd experiment show the sme shift direction of the nturl frequencies while the pper is geing the sensitivity of the model is much less. The most likely reson for these differences is the limittions of model itself. The winding discs re mde s spirl type but this preliminry model uses concentric circulr rings. Whether this method cn fully describe the mechnicl chrcteristics of the winding disc or not needs more reserch. nother limittion is the pper is only modelled in its out-of-plne direction (the rdil direction of the disc). However the in-plne vibrtion of the disc is determined by both rdil force nd tngent force which mens the pper should be modelled in its in-plne direction s well to 5. CONCLUSIONS ND FURTHER WORK geing of winding insultion pper is one of the most serious power trnsformer filures. This pper provides some experimentl nd theoreticl evidence supporting tht the vibrtion chrcteristics of the winding disc is ffected by the mechnicl properties of the insultion pper tht vry with geing extent. Specificlly when the pper ges the mechnicl properties of the pper in its out-of-plne direction decrese. s result the nturl frequencies of winding disc shift to low frequency end. This reserch revels potentil possibility tht the geing extent of winging insultion pper cn be detected by monitoring the winding s mechnicl vibrtion. The further work of this reserch will focus on two spects. One is improving the ccurcy of the existing concentric circulr rings model. The difference between the spirl disc nd concentric rings will be simulted by FEM s it is difficult to chieve this experimentlly becuse of the inevitble errors such s the geometricl error of the tested items due to the mnufcture procedure. Menwhile more ccurte rings model will be developed by introducing the pper effect in the tngentil direction of the disc. The other spect of the further reserch is building multilyer disc model tht is closer to ctul disc-type winding. Experiments to nlyse some other winding filure modes such s loss of winding clmping pressure nd winding prtil deformtion will lso be performed. CKNOWLEDGEMENTS We thnk Dr. ndrew Guzzomi for proof reding the mnuscript nd useful suggestions. Finncil support to this project by CRC for Infrstructure Engineering sset Mngement (CIEM) is grtefully cknowledged. REFERENCE [1] M. Wng et l. Review of condition ssessment of power trnsformers in service IEEE Electricl Insultion Mgzine Vol 18 No. 6 pp [] CIGRE Working Group 5 n interntionl survey on filures in lrge power trnsformers in service Electr 198 No. 88. [] B. Grcí et l. Winding deformtions detection in power trnsformers by tnk vibrtion monitoring Electric Power System Reserch 74 5 pp [4] S. C. Ji Filure Monitoring Investigtion of Trnsformer Wingding nd Core Chrcteristic Xi n Jio Tong University. [5] Z. Berler et l. Vibro-coustic Method of Trnsformer Clmping Pressure Monitoring Conference Record of the IEEE Interntionl Symposium pp [6] T. K. Sh nd P. Purkit Understnding the impcts of moisture nd therml geing on trnsformer s insultion coustics 11 7

8 -4 November 11 Gold Cost ustrli Proceedings of COUSTICS 11 dielectric response nd moleculr weight mesurements IEEE Trnsctions on Dielectrics nd Electricl Insultion 8 Vol. 15 N. pp [7] D. J. T. Hill study of degrdtion of cellulosic insultion mterils in power trnsformer prt 1. Moleculr weight study of cellulose insultion pper Polymer Degrdtion nd Stbility pp [8]. E. H. Love Tretise on the Mthemticl Theory of Elsticity" New York Dover Publictions 197. [9] T. Chrnley et l. Vibrtions of thin rings of rectngulr cross-section Journl of Sound nd Vibrtion 1989 Vol. 14 Issue pp [1] V. I. Lzrev nd V. E. Rushchk Equtions of circulrring oscilltions Plenum Publishing Corportion 198 pp coustics 11