44. Creatng of the mathematcal model of a reslent support sprng type element and ts transfer functon from the nput and output mpulse responses V. Slvnskas, K. Slvnskas, A. Trumpa Vlnus Gedmnas Techncal Unversty Basanavcaus 8, LT-034, Vlnus, Lthuana e-mal: astera@astera.lt, Kastyts.Slvnskas@me.vgtu.lt, Andrus.Trumpa@me.vgtu.lt (Receved 08 January 009; accepted 0 March 009) Abstract. Developng of the transfer functon of the sprng type element of the reslent bearng support by creatng adequate mathematcal models of processes for partcular frequency ntervals from sgnals obtaned as mpulse responses of the sprng at ts nput and output measured n the stand under mpact at the nput s analysed n the artcle. Impulse responses are nvestgated by the Fourer method by separatng components of partcular frequences. Mathematcal models of the processes at the sprng nput and output are developed as the sum of models-formants correspondng to partcular frequency ranges. Formants themselves are modeled by the sum of damped sne waves. Parameters of the model are estmated by the teratve Levenberg method. The mathematcal model developed further s used for dagnostcs of support bearngs that are nsulated from the external body by these reslent elements. Keywords: mathematcal model, falure dagnostcs, reslent support, exctaton, transfer functon, frequency response Introducton Non-destructve dagnostc researches of the system for the evaluaton of ts operatng state sutablty or detecton of sgnals of ntal falures allowng to use preventve means to avod them are wdely spread n practce and are very effectve. They are wdely used for the evaluaton of the sutablty of hgh-speed rotor system bearngs wthout stoppng ther operaton and dsassembly. Although there are systems n whch such research s complcated due to constructon partculartes. One type of such systems are centrfugal systems whose bearng supports n most cases are connected wth the external frame through reslent elements restrctng the transmsson to the frame of all row addtves of the spectre of oscllaton caused by bearng falures [, ]. For a proper evaluaton of transfer propertes of such a system t would be expedent to research the reslent element transfer functon by creatng ts adequate mathematcal model n frequency ranges whch are expected to cover possble frequences caused by bearng falures. For ths am, t would be expedent to determne the transfer functon of the reslent element. Ths would allow to make a decson about the possblty of fndng frequences caused by the bearng falure by measurng oscllatons of the external body. The sprng element s a reslent element used n a centrfugal mlk separatng apparatus the dagnostc research of whch s an mportant mean for a full and safe usage of the allowable workng resource of t. It was chosen as the research object. 49
Expermental equpment The experment was carred out on the research stand shown n Fg.. The sprng element was trapped between two reslent systems, nsulated one from the other. Fg.. Stand for researchng of oscllatons transfer propertes of the sprng: the pezoelectrc accelerometer; the sprng, 3 the elastc band The stand was composed by two massve plates wth comparatvely not rgd bands of rectangular cross secton fxed to them. They mtate two separated systems of the centrfugal mlk separatng apparatus the shaft and the body. These systems were nsulated one from the other by reslent rubber supports. Measurng sensors were pezoelectrc accelerometers, fxed to both ends of the sprng, and they regstered acceleratons of vbratons of both reslent systems at the ponts of the sprng ends. Mostly, bearng falures cause a pulse type exctaton accompaned by wde frequency spectre oscllatons the appearance of whch s the base for the dagnoss of bearng falures. So, a seres of mpacts was used as the exctaton gven to one of the reslent systems and transent processes (tme sgnals) of both ends of the sprng were measured. The response of one system to the exctaton and ts transfer to the other system was researched. Research procedure In order to research oscllatons transmttng propertes of the sprng, t s necessary to create ts adequate mathematcal model allowng to calculate ts transfer functon, by the help of whch t would be possble to estmate the nfluence of the reslent element to proper bearng dagnostcs. In order to create the mathematcal model, a seres of blows was submtted to the upper plate. The obtaned nput and output sgnals are shown n Fg.. Realzatons n the form of seres of mpacts allow to choose a better qualty sgnal to be researched. The level of the output sgnal of the sprng was approxmately 0 tmes weaker than the nput sgnal. The standard voce sgnal dgtsng the procedure was used to dgtse these analogcal sgnals obtaned from accelerometers. Then the sgnals were fltered from addtves of hgher than 5000 Hz and were ntegrated twce. The realzatons obtaned were processed by the FFT procedure, and spectres of frequences of the nput and output sgnals were analysed. The spectrum of the measured sgnals s very wde, and t s dffcult or even mpossble to model 50
such a process n a ratonal form for all frequences. The spectres of nput and output sgnals of the sprng are shown n Fg. 3. Fg.. Input (upper) and output (bellow) sgnals measured at ends of the sprng a) b) Fg. 3. Frequency spectres of the nput (a) and output (b) sgnals However, bearng falures cause oscllatons of partcular frequences whch are n partcular ranges determned by the rotatonal frequency of the bearng, the number of rollng elements n the bearng, bearng work condtons [, ]. The man characterstc frequences related wth the bearng falures of the mlk separatng apparatus rotor system consstng of an nvestgated sprng element are the followng []: 46,4 Hz (the rotaton frequency of the bearng retaner),,5 Hz (the rotor rotaton frequency), 5,0 Hz (the doubled rotor rotaton frequency), 60,5 Hz (the frequency of the run of balls by the outer race), 65,8 Hz (the rotaton frequency of the ball), 859,7 Hz (the frequency of the run of balls by the nner race), 05,5 Hz (the doubled frequency of the run of balls by the outer race), 3,5 (the doubled rotaton frequency of the ball). 5
We found that for modellng of nput and output sgnals, t s better to take a short frequency nterval. The chosen nterval should cover partcular frequences, whch mght occur due to the bearng partcular falure. It s necessary to observe that sometmes the spectre of the partcular frequency nterval created n the stand upper system does not have a suffcent level of exctaton ampltudes. Ths s the nput exctaton for the sprng. In ths case the dynamc characterstcs of the stand upper system (stffness or mass) should be changed to obtan some level of the nput sgnal. So, we attempted to model nput and output processes n the followng ntervals of frequences: 40-55 Hz, 0-90 Hz, 00-50 Hz, 480-600 Hz, 600-690 Hz, 795-885 Hz, 5-340 Hz, 04-080 Hz. Frequency spectres for one of the nvestgated short frequency ntervals of the nput and output sgnals are shown n Fg. 4. a) Fg. 4. Spectres of the nput (a) and output (b) sgnals of the frequency nterval 600-690 Hz b) The modellng of these sgnals was made followng [3-5]. Modellng of the sprng nput and output sgnals We performed the modellng by creatng a mathematcal model n the form of the sum of damped nteractve sn waves. The sprng mpulse response of both, the nput and output, was modelled as a sum of formants quas-polynomals n the form: h( t ) = m = f ( t ), () where polynomals are composed from polynomals components havng multplers calculated from damped snusods of a partcular frequency [5]. We restrcted the polynomal tll the quadratc order: 5
λ f ( t ) = e t [ a + a t sn( πω t+ sn( πω t+ ) + a 3t ) + sn( πω t+ 3 )], () where ω s the angular frequency of the searchng formant, λ s ts dampng factor, a a 3 are ampltudes of the searchng formant components and φ φ 3 are component phases. The estmaton of frequences and dampng factors was performed by usng the Prony method, and for mprovng the ntal estmated values the teratve non lnear Levenberg optmsaton was used followng [3]. The algorthm of calculaton of polynomal members parameters s descrbed n detal n [3]. One of the partculartes durng modellng processes for the short frequency nterval s the acceptaton of the number of formants composng the model. We fnd that for a short frequency nterval n our case s expedent to take one or two formants (m= or m=) correspondng to frequences close to those seen on the ampltude-frequency characterstc of the gven frequency nterval. The RSME evaluatng the correctness of the model n such a case dd not exceed 5% for all frequency ranges. Calculated parameters of members of polynomals modellng nput and output sgnals n the frequency nterval 600-690 Hz are presented n Table. TABLE : Parameters of model components for the frequency nterval 600-690 Hz Sgnal Input (before sprng) Output (after sprng) Frequency, ω, Hz Damp., λ, /s 6-67 676-76 6-69 673-84 Ampl., a, unts Phase, φ, rad 654,6544 3,0749,949 -,66 0,00 0,8889 974,633 0,857,67-0,84 0,003 -,673 5,937,587 0,003,9503 0,0,055,644 0,569 0,05 -,4 0,0,406 Components representng three addtves of the frst polynomal, whch models the nput process, are shown n Fg. 5. The nput process for the frequency nterval 600-690 Hz was modelled by usng two polynomals formants. The modelled and real oscllaton processes at the sprng nput are shown n Fg. 6. We found that the modellng of the oscllaton process s more effectve by usng a small tme nterval sgnal taken at the begnnng of the oscllaton process, because all frequences are expressed more clearly and can be sngled out due to more powerful oscllatons. Durng a long tme nterval some frequences damp, the non-lnearty of the system also results n changes of the process character, and the modellng of the long tme nterval process leads to a complcated model. Thus, for the modellng of the process of all researched frequency ntervals we used the tme nterval of duraton 0,008-0,0435 s. 53
Fg. 5. Components of the frst polynomal of the sprng nput process model for the frequency nterval 600-690 Hz ` a) b) c) d) Fg. 6. The frst (a), the second (b) formants, the model the sum of these formants (c) of the oscllaton process, and the actual oscllaton process (d) for the frequency nterval 600-690 Hz Determnaton of the sprng transfer functon We calculated the transfer functon n the followng way. The Fourer transform of each formant can be expressed as [5]: 54
a e a e h ( f ) = j ( λ πj( f ω )) ( λ πj( f + ω )) a e j ( λ πj( f ω )) j a e 3 3 j ( λ πj( f ω )) j j 3 a e ( λ πj( f + ω )) j a e 3 3 ( λ πj( f + ω )) j j where s the formant number. The Fourer transform of the modelled sgnal wll be the sum of Fourer transforms of all m formants, composng the gven sgnal,.e., h ( f ). = The sprng frequency response, H(jw), s calculated as the rato of the Fourer transforms of the output sgnal and the nput sgnal H( jw ) = moutput output = mnput = h h nput ( jw ) ( jw ) 3, (4) where w= πf and t s the angular frequency. The range of frequences, for whch the Fourer functon s determned, should be narrower to some extend than the range of modellng n order to avod dstortons related wth the boundary dvergences. After dvdng the Fourer functon of the output sgnal by the Fourer functon of the nput sgnal, we wll get the transfer functon for the frequency range, for whch these sgnals were determned. Ampltude-frequency responses of the sprng nput and output for the frequency ntervals 480-60 Hz and 600-690 Hz are shown n Fg. 7. The transfer functon of the sprng n the frequency nterval 560-680 Hz, whch s composed of the transfer functons calculated n two frequency ntervals: 480-60 Hz and600-690 Hz s shown n Fg. 8., (3) a) 55
b) c) d) Fg. 7. Ampltude-frequency responses of the sprng nput (a, c) and output (b, d) for frequency ntervals 480-60 Hz and 600-690 Hz Fg. 8. The transfer functon for frequency nterval 560-680 Hz It s seen (Fg. 8), that transfer functons of two frequency ntervals calculated separately gve a rather good correspondence n ther touchng pont at 605 Hz. The transfer functon of the sprng n all frequency ntervals related wth the appearance of possble bearng falures frequences s shown n Fg. 9. 56 Fg. 9. The transfer functon for all nterestng frequency nterval 0-300 Hz
Conclusons The sprng transmts the nput sgnal. The transfer functon of the sprng can be used to dagnose the bearng falure by analysng the change of the spectre of frame vbratons durng the tme and makng decsons about reasons of frequences appeared. However, the output sgnal of the sprng n most nvestgated frequency ntervals s approxmately 0 tmes weaker than the nput sgnal. The transfer functon s more expedent to create for small frequency ranges, n whch frequences caused by partcular bearng falures can appear. The model constructed for such a case s more adequate and the error of the bas from the measured sgnal s less. In order to determne the bearng falure frequency at the sprng output (mlk separator outer body) to fnd the bearng falure frequency n the spectre of the body oscllatons, t s necessary to fx and carefully analyse the oscllaton level of the outer body wth the rght bearng and compare t wth the current body oscllaton level. References [] Vekters V., Trumpa A., Cereska A.. The dagnoss of the trbologcal systems wth vbraton nsulaton. Journal of Vbroengneerng. Volume 5, Nr. (0), 003, p. -3. [] Barzdats V., Čnkas G.. Montorng and Dagnostcs of Rotor Machnes. Kaunas, Technologja, 998, 366 p. (In Lthuanan). [3] Slvnskas V. and Šmonyt V. Estmaton of Parameters of Impulse Responses of Mechancal Systems by Modfed Prony Method. Volume 3 of Sold State Phenomena. Mechatronc Systems and Materals. Trans Tech Publcatons LTD. Swtzerland, 006, p. 90-94. [4] Slvnskas V., Slvnskas K., Bučnskas V. Creatng a mathematcal model from the Realzaton of the Measurng of a transent Process of a Mechancal System. Volume 3 of Sold State Phenomena. Mechatronc Systems and Materals. Trans Tech Publcatons LTD. Swtzerland, 006, p. 9-34. [5] Slvnskas V., Šmonyt V. Mnmal Realzaton and Formant Analyss of Dynamc Systems and Sgnals. Vlnus Mokslas, 990. Reprnted by Booksurge, 007. 68 p. (n Russan). 57