Regional Stability Analysis of Rotor-Ball Bearing and Rotor- Roller Bearing Systems Considering Switching Phenomena

Transcription

1 ternational Journal of Mehanial, Aeropae, dutrial, Mehatroni and Manufaturing Engineering Vol:6, o:3, Regional Stability Analyi of Rotor-all earing and Rotor- Roller earing Sytem Conidering Swithing Phenomena Jafar Abbazadeh Chekan, Kaveh Merat, Haan Zohoor ternational Siene dex, Mehanial and Mehatroni Engineering Vol:6, o:3, waet.org/publiation/3888 Abtrat thi tudy the regional tability of a rotor ytem whih i upported on rolling bearing with radial learane i tudied. The rotor i aumed to be rigid. Due to radial learane of bearing and dynami onfiguration of ytem, eah rolling element of bearing ha the poibility to be in ontat with both of the rae (under ompreion) or loe it ontat. A a reult, thi hange in dynami of the ytem make it to be known a withing ytem whih i a type of Hybrid ytem. thi invetigation by adopting Multiple Lyapunov Funtion theorem and uing Hamiltonian funtion a a andidate Lyapunov funtion, the tability of the ytem i tudied. The purpoe of thi tudy i to inpet the regional tability of rotor-roller bearing and rotor-ball bearing ytem. Keyword Stability analyi, Rotor-rolling bearing ytem, Swithing ytem, Multiple Lyapunov Funtion Method I. ITRODUCTIO OTORDYAMICS i an important field of tudy in R dynamial ytem. eaue of demand for rotary mahine with high peed uh a turbine, pump, fan, pindle mahine and o on, the dynami and tability analyi of thi type of ytem are onidered over a hundred year. A ome of the rotor ytem are upported by rolling bearing with radial learane, the tability analyi of uh ytem an be helpful in the orreponding area of the indutry. The dynami equation of rotor-bearing ytem are motly ompliated with high nonlinearity; a a reult, the analytial olution for thee equation i ophitiated and in ome ae i impoible to obtain; therefore, it i neeary to ue the tability riteria uh a Lyapunov theory, Routh- Hurwitz method or Floquet theory to analyze the tability of rotor-bearing ytem. The bearing play an intrumental role in the dynami of rotor bearing ytem. Hene, invetigation are done in analyzing the effet of the bearing parameter uh a tiffne and damping oeffiient on the tability region of the rotor-bearing ytem. [] tudied the effet of bearing parameter on the tability of the rigid rotor ytem. their tudy they applied Routh-Hurwitz riterion to tudy the J. Abbazadeh Chekan, Department of Mehanial Engineering, Sharif Univerity of Tehnology, Tehran, Iran. ( afar.abbazadeh@gmail.om). K. Merat, Department of Mehanial Engineering, Sharif Univerity of Tehnology, Tehran, Iran. ( k_merat@meh.harif.edu). H. Zohoor, Prof., Department of Mehanial Engineering, Sharif Univerity of Tehnology, Tehran, Iran. (orreponding author to provide phone: ; fax: ; zohoor@harif.edu). influene of prinipal and ro-oupling tiffne and damping oeffiient on the tability region of the ytem. addition, [] tudied the tability of a rigid rotating haft, upported by bearing idential to ref [] but uing the Lyapunov riterion, and obtained the imilar reult to their previou work. Furthermore, in another invetigation [3] tudied the tability of a rotor ytem whih inlude elati haft and i upported on the bearing with nonlinear parameter. Lyapunov riterion i employed in hi invetigation and the Hamiltonian funtion i onidered a Lyapunov andidate funtion. Finally, the influene of haft parameter uh a Yung Elati modulu and bearing parameter are tudied in [3]. the above mentioned reearhe the bearing are aumed a ideal boundary ondition uh a imple pring and damper whih are not appropriate model for rolling bearing. a work by [4], a more preie mathematial model of rolling bearing i developed. their work the bearing i modeled by Hertzian ontat theory for eah of rolling element, onidering the radial learane, the bearing fore and moment. ome other tudie uh a [, 6] the nonlinear dynami of a rotor with ball bearing having radial learane i alo invetigated. A the rotor ytem with the rolling bearing whih have radial learane exhibit withing behavior aoiated with the ontat of the rolling element with the rae of the bearing, it fall into the ategory of hybrid ytem. The hybrid ytem ombine ontinuou dynami, repreented by differential or differene equation, with finite tate dynami uually alled finite automaton. A peial la of thee ytem i withed ytem whih are onidered in our model. The withed ytem model onit of finite differential equation ome rule deignate whih one i the governing differential equation at any time interval. So in order to invetigate the tability of thi bearing model, one hould extend tability theorem for withed ytem whih are briefly diued below. The tability of withed ytem ha been widely been invetigated, but only ome of the more relevant are ummarized. [7, 8] introdued Multiple Lyapunov funtion to tudy the tability of withed ytem. [9] Hepanha tudied uniform tability for linear withed ytem by extending the LaSalle' variane Priniple whih i alo baed on Multiple Lyapunov funtion method. Alo finding one ommon Lyapunov funtion for proof of tability i implemented in [] by uing Lie-Algebra. The tability of withed linear ytem having diipative Hamiltonian are alo ternational Sholarly and Sientifi Reearh & novation 6(3) 68

2 ternational Journal of Mehanial, Aeropae, dutrial, Mehatroni and Manufaturing Engineering Vol:6, o:3, ternational Siene dex, Mehanial and Mehatroni Engineering Vol:6, o:3, waet.org/publiation/3888 invetigated under ertain withing equene related to power-onverter with ideal withing in Gerriten, K., A. Van der Shaft, and W. Heemel invetigation []. Similarly Zhu, L. and Y. Wang [], inpet the tability of withed diipative Hamiltonian ytem under arbitrary withing path. oth artile [, ] hoe their diipative Hamiltonian funtion a their repetive Multi-Lyapunov funtion to onider the aymptoti tability of ytem. Our tability proof i imilar to thoe in [, ] but lightly different beaue of lak of diipation in our ytem whih lead to invetigation of regional tability for uh ytem. Here, the tability region i defined a a region that ytem tate never leave if the initial ondition i in predefined et. thi tudy the rotor inlude a haft whih i aumed to be rigid and i upported on two rolling bearing (either roller bearing or ball bearing). Alo, in dynami modeling, it i aumed that the ytem ha two tranlational Degree of Freedom (-DOF). The mathematial formulation i preented in Setion II. Setion IIII by adopting the Multiple Lyapunov Funtion theorem and uing Hamiltonian funtion a a andidate Lyapunov funtion, the regional tability of the ytem i tudied onidering withing in ytem dynami. Furthermore in Setion IV the effet of load-tiffne parameter and rotor peed on regional tability of roller bearing and ball bearing will be invetigated, and ome numerial olution are preented to how the relation between the olution traetory and the dedued regional tability. II. PROCEDURE FOR PAPER SUMISSIO A it an be een in Fig., the rotor ytem under tudy ontain rigid haft whih i upported by two rolling bearing. Here, two ae are onidered for modeling, in the firt ae the ytem bearing are onidered to be ball bearing type and in the eond ae, the bearing are aumed to be roller bearing. Fig. The hemati diagram of rotor bearing ytem Some aumption whih are inluded in thi tudy are: the haft i onidered to be rigid, traight, ymmetri and balane, the bearing are uppoed to be fritionle. Furthermore, it i aumed that the haft ha two tranlational degree of freedom in x, y diretion depited in Fig.. Firt, in thi etion the kineti energy term of the ytem will be evaluated and at the next tep the elati deformation energy of both type of bearing will be alulated, to extrat the Hamiltoniann Funtion of rotor bearing ytem. A. Kinemati Analyi thi tudy two oordinate ytem are onidered. The firt oordinate ytem, XYZ i global oordinate ytem whih i loated at ma enter of the rotor and it Z axi i along with the enter line of the bearing outer rae and haft rotate with rotation peed Ω around the Z axe. The eond oordinate ytemm xyz i onidered to be attahed to the rotor at the ma enter point and it unit vetor are introdued by i,, and k. eaue of radial learane and elati harateriti of the rolling element of bearing, the haft an have diplaement in perpendiular diretion, o thi diplaement x and y are onidered to be along i, diretion o, thee poition of ma enter of rotor, r, an be written a: r xi y () r Differentiatingg () with repet to time, the veloity vetor i r : x yi y x () y onideringg that m i the ma of the rotor and I p i the rotor moment of inertia along the Z axe, the kineti energy of the ytem ann be omputed a: T m x y y x I p (3). Modeling earing thi tudy, the ytem i modeled with two type of bearing; roller bearing and ball bearing. The hemati figure of the bearing are hown in Fig., Fig. how the angular poition of -th rolling element for both type of bearing,, Fig. (b) and () repreent the elati deformation of -th roller element in the roller bearing and ball element in the ball bearing repetively. order to obtain potential energy of the ytem, Hertzian ontat theory i applied. y onidering the kinemati hown in Fig. the angular poition of -th rolling element,, for both type of bearing an be obtained by: t,,... (4) in whih repreent the number of rolling element and i the rotation peedd of bearing age and i [, 6]: ri A, A () r r i o r i and r o are inner rae and outer rae radiu. Due to relative diplaement of inner and outer rae of bearing, eah rolling element of bearing an have elati deformation. thi tudy the outer rae i fixed on a rigid upport and inner rae i fixed on the haft, o it ha the ame diplaement a the haft. So, the inner rae diplaement i: xm xco t ysint (6) ym x Sin t ycot xm and ym are inner rae diplaement in global oordinate ytem. The effetive diplaement of -th rolling element for both roller bearing [4] and ball bearing an be written a[, 6]: ternational Sholarly and Sientifi Reearh & novation 6(3) 68

3 ternational Journal of Mehanial, Aeropae, dutrial, Mehatroni and Manufaturing Engineering Vol:6, o:3, ternational Siene dex, Mehanial and Mehatroni Engineering Vol:6, o:3, waet.org/publiation/3888 Co r xm ymsin (b) () Fig..Deription of the bearing, loation of the -th rolling element, (b) The elati deformation of the -th ball element, () The elati deformation of the -th balll element Subtituting Eq. (4)-(6) into Eq. (7), we have: r x Co t ysin t (8) in whih ( A ) order to omit the expliit term of time from the ytem formulation for tability analyi, the oordinate tranformation i onidered a: x XSin tycot (9) y X Co ty Sint If above oordinate tranformation i applied to the ytem, the term T andd r an be rewritten in new oordinate ytem a: T m X Y A( XY YX ) A ( X Y ) I () p r XSin YCo y onideringg Fig.() for a roller bearing the deformation of roller element i: R r R R () R R above equation how that for R the -th roller element i not under ompreion. y adopting imilar ondition for ball element, the -th ball element elati deformation an be written a: t R (7) elati r () y applying Hertzian ontat theory for -th rolling element (for both of roller and balll element) the orreponding reultant normal load an be aumed a: ere n Q K n (3) whe an be either or R. The elati deformation energy of -th rolling element ould be written a: Kn n U n (4) y applying uperpoition prinipal on all of rolling element, the potential energy of the ytem, U, i derived from Eq. (4) a: Kn n U () n Due to imilarity of both left and right bearing in the ae tudie (rotor-roller bearing and rotor-ball bearing ytem), it an be onludedd that the potential energy for both of rotor bearing ytem i: Kn n U (6) n y ubtituting Eq. (6)-() into Eq. (6), and onidering exponent n i equal to 3/ for ball element with elliptial ontat and i / /9 for roller type with retangular ontatt [4], the potential energy for the firt ae tudy with roller bearing, U R, and the eond one with ball bearing, U, are: 9 9 UR 8 R 9 (7) K U 4 and K are effetive tiffne ontant for roller element ontatt and ball element ontat repetively. III. REGIOAL STAILITY AALYSIS thi artile by onidering bearing radial learanee and dynami onfiguration in both of the rotor ytem, eah rolling element an be ompreed between the rae (turn into ative rolling element) or loe it ontat and beome tre free element (turn in to non-ative rolling element). Hene, onidering hange in ontat tatu of roller element and onequently, the hange in the dynami equation of ytem thee rotor ytem ould be ategorized a withing ytem. other word the ytem ha finite different dynami a: x fit x i t (8) n x are the tate of the dynamial ytem, the index i(t) repreent that ytem i in whih mode of ontat and thi index i hanged with repet to ontat ondition defined in ternational Sholarly and Sientifi Reearh & novation 6(3) 68

4 ternational Journal of Mehanial, Aeropae, dutrial, Mehatroni and Manufaturing Engineering Vol:6, o:3, ternational Siene dex, Mehanial and Mehatroni Engineering Vol:6, o:3, waet.org/publiation/3888 (), () and indiate the number of all poible mode of withing in the ytem that an our. A a reult of geometri arrangement of rolling element in the bearing, and due to the radial learane of bearing the maximum number of rolling element whih ould be in ontat (under ompreion) i braket repreent the floor funtion. Hene, repreent the number of all ubet of I={,, } whih have at mot member. When the ytem i in the i(t) mode, we an ue Hamilton Funtion a an eligible Lyapunov Funtion. Sine the ytem i a Hamiltonian ytem, Hamilton Funtion in eah mode an be written a: H T U T (9) in whih the ubript of T denote the degree of homogeneity in the generalized veloity variable, q i. Due to fritionle aumption of roller element ontat, there i no damping in thi dynamial ytem. A a reult, in eah mode of ytem, i(t), and when no withing our, the value of Hamilton Funtion i ontant [,3], i.e., H. It i notable that at the intane of ontat (withing time), there i no ompreion on the rolling element, o there will be no hange in the potential energy in withing. Alo in mall infiniteimal time withing happen, the kineti energy will remain the ame. Hene at withing intant the value of Hamilton Funtion doen t ump and remain the ame. Therefore, it an be onluded that the Hamilton value of thi hybrid ytem remain ontant in the total time interval inluding all the withing. Thi property will be later ued in tability analyi. After eah withing phenomena in the ytem, the number of ative rolling element (rolling element under ompreion) ould alter; a a reult of thee hange, different dynami mode will be ativated. The Multiple Lyapunov Funtion approah, propoed in [8], i applied here for tability analyi a delared in the following theorem: Theorem Suppoe there are andidate Lyapunov funtion Vi ( x), i,...,, for repetive vetor field x fi( x), and eah Vi ha following propertie. For eah poible x in mode i, there i a real ontant, that Vi ( x),and the boundedne of Vi ( x ) hould lead to boundedne of tate, x. All Vi are non-inreaing in their orreponding mode [ V i( x) ] V V at eah withing time 3. i( t ) i( t ) Then, it an be inferred the withing ytem i table in Lyapunov ene. Hamilton funtion in eah mode i(t) i an appropriate Lyapunov funtion V i. A it wa mentioned before hoen Lyapunov funtion (Hamilton Funtion) have no hange at the withing time; i.e, a moment before and after the with, V H H V, therefore third ondition i i( t ) i( t ) i( t ) i( t ) atified. Alo, the eond ondition i fulfilled due to ontant value of Hamilton Funtion in eah mode of ytem a mentioned before. Therefore, the only remaining ondition that aure the tability of the ytem i the firt ondition. Regarding the Hamilton funtion in (9), the term T i dependent on generalized veloity and ( U T ) depend on generalized oordinate, o in order to verify the firt riteria, the term T and U T are analyzed eparately. Due to ontant value of Hamilton funtion, whih i equal to initial value, H, and the poitive definitene of term T ( T ), we ould write: U T H T H () So, it an be inferred that the term U T i bounded from above. Alo beaue of higher power of generalized oordinate in poitive definite potential term U with repet to entrifugal termt the lower boundne of the term U T i eaily onfirmed; U T u u. So the term ( U T ) i ompletely bounded from above and below. Furthermore ( U T ) i a non-frational funtion in poltnmial format, whih lead to boundedne of generalized oordinate that omprie loed region in (X, Y) domain. It i obviou that ytem annot leave the mentioned region. So thi region, namely; S i alled the tability region of the ytem in (X,Y) domain. It i notable that the boundary of region S ould be found by ondition: U T H () Alo the upper bound on T ould be obtained from the lower bound of the term U T a: T H u () Conidering poitive definitne and upper boundne of T, it an be inferred that, generalized veloitie are alo bounded. So if the ondition () of Theorem hold, the ytem will be table and all tate remain in ertain region, in partiular the diplaement of rotor i onfined in S. For further diuion of thi type of tability, the following definition i given: Definition of -region table: A ytem i -region table if S C( ), C( ) i quare region in X-Y plane with width and it enter i at origin. The term U T an be written a: U T I (3) p the term for roller bearing and ball bearing an be written a: 9 9 R 8 R ma ( X Y ) (4) 9 ternational Sholarly and Sientifi Reearh & novation 6(3) 683

5 ternational Journal of Mehanial, Aeropae, dutrial, Mehatroni and Manufaturing Engineering Vol:6, o:3, ternational Siene dex, Mehanial and Mehatroni Engineering Vol:6, o:3, waet.org/publiation/3888 K 4 ma ( X Y ) () R and are related to the rotor ytem whih i upported by roller bearing and ball bearing repetively. the imilar way, the term H an be rewritten a: H I ( X, Y, X, Y ) (6) p for rollerr bearing, (XX, Y, X, Y ) i defined a: 9 ( X, Y, X, K R 9 Y ) 8 R 9 X X Y Y (7) m ( X Y ) ma ( X Y ) and for ball bearing, we have: ( X, Y, X, K Y ) 4 X X YY (8) m ( X Y ) ma ( X Y ) Hene, without any lo of generality, the expreed ondition in Eq.() an be expreed a: ( X, Y, X, Y ) (9) whih mean the boundary of S i dependent on the initial value of ( X, Y, X, Y ), whih itelf dependent on initial diplaement and veloitie of dynamial ytem. Depending on initial ondition, ( X, Y, X, Y ) ould take poitive or negative value. ut for analyi in the ret of artile the value of equal zero i taken a a referene. thi ae H beome I p. IV. RESULTSS AD DISCUSIOS thi etion, regional tability of rotor-roller bearing ytem and rotor-ball bearing ytem are tudied numerially. Alo the effet of load-tiffne parameterr and rotation peed on the regional tability of ytem for both type of bearing will be inpeted. The peifiation of roller and ball bearing are deribed in Table. Due to retrited defletion apaity of eah rolling element, in thi tudy the generalized oordinate are onidered to have the maximum value of n (n i real poitive number whih i hoen onidering the deired region of tability). So, by onidering thi aumption, the funtion R and are plotted with repet to the diplaement (x, y) within thi domain. It i notable that in thi ytem the withing phenomenon take plae frequently, a a reult in eah diplaement the number of ative rolling element alter. For intane when the generalized oordinate are in the learane area, X Y, there are no rolling element under ompreion and the term R and ha negative value in thi region, o the generalized oordinate x and y are nlined to inreae. ut out of thi learane area the rolling element ome in ontat gradually and the value of R and inreae aordingly. It i aumed that the radial learane value and inner and outer rae radiu are the ame in both type of bearing whih upport the ame rotor. Parameter r i r m TALE I DESIG PARAMETERS FOR TYPICAL ALL AD ROLLER EARIGS Deription earing inner rae radiu earing outer rae radiu Radial learane Rotor ma Value 39(mm) (mm) μ Kg For roller bearing, the term i plotted in Fig.3 for n=3. The reult alo learly emphaize that due to higher power of generalized oordinate in the term U with repet to T, the ytem i table. Fig.3 Surfae of R for rotor- roller bearing ytem in the ae of ( X, Y, X, Y ) ; rad /, 4, 7 /9.6 / m Taking into aount -region table definition whih i highlighted before, -region tability of the ytem i invetigated for both of the rotor-roller bearing and the rotor- balll bearing ytem whih i illutrated in Fig.4. Thee figure how the effet of load-tiffne parameter ( K R, K ) on deired regional tability ( ) of the ytem. A it an be een in Fig.4 and Fig.4(b), inreaee in the value of K R, K, make ytem to reah maller tability region (whih mean lower amplitude in oillation of rotor). For 8 / 9 intane with / m for different initial ondition whih atify ( X, Y, X, Y ) or H / I p, the ytem would remain in it orreponding region,. ut a K R dereae the region enlarge e in 6 /9 K R 4 / m the ytem in no more -region table. R μm ternational Sholarly and Sientifi Reearh & novation 6(3) 684

6 ternational Journal of Mehanial, Aeropae, dutrial, Mehatroni and Manufaturing Engineering Vol:6, o:3, ternational Siene dex, Mehanial and Mehatroni Engineering Vol:6, o:3, waet.org/publiation/3888 (b) Fig. 4 Effet of load- tiffne parameter of rolling element ontat on tability region, Γ, for ; a) rotor-roller bearing ytem, b) rotor-ball bearing ytem. For both ae 6, rad / A it i obviou from Eq. (4) and ( ) the inreae in rotation peed will enlarge the region thatt ytem remain in. Conidering the definition of. tability, the effet of rotation peed on regional tability of both rotor-roller bearing and rotor-ball bearing ytem are inpeted and hown in Fig.. Aordingly, the reult how by inreae in the value of rotation peed of the rotor, the region ; would grow up whih mean higher amplitude in oillation of rotor. Fig. Effet of rotation peed of rotor on tability region, Γ, for both roller bearing and ball bearing ae, 6 / / m, K 4 / m 3/, 4 order to validate the tability reult, the nonlinear dynami equation of motion for ytem are olved, numerially. y utilizing Lagrange method the equation of motion for ytem an be written a: m X AmY m A my AmX m A Y K Co Above equation are olved for two different initial ondition whihh atify ( X, Y, X, Y ) and an initial ondition ( X, Y, X, Y ). Reult are hown in X-Y plane, illutrating the enter of ma movement (ee Fig.6). Alo the tability region of ytem; whih i the reult of regional tability analyi of ytem ha given in Fig..6-6() for omparion. A it an be een in Fig. 6 and 6(b) for ( X, Y, X, Y ), the traetory of enter of ma X K Sin / 9 R R / 9 R R ome loation beome tangent with the border of. A it i expeted, for initial ondition ( X, Y, X, Y ), the traetory of ytem i ompletely inide the obtained tability region, admittedly Fig. 6() depit thi predition a well. remain in it orreponding tability region,, at (3) ternational Sholarly and Sientifi Reearh & novation 6(3) 68

7 ternational Journal of Mehanial, Aeropae, dutrial, Mehatroni and Manufaturing Engineering Vol:6, o:3, ternational Siene dex, Mehanial and Mehatroni Engineering Vol:6, o:3, waet.org/publiation/3888 Fig. 6 Diplaement of rotor-roller bearing ytem( 4, 7 / 9 K R.6 / m, rad / ) in X-Y plane under different initial ondition for; a) ( X, Y, X, Y ) e X Y m, X Y 3.47 m/ ; b) ( X, Y, X, Y ) e X Y m X Y m/ ; ) ( X, Y, X, Y ) -.38 X, Y m, X Y m/ (b) () REFERECES [] A. El-Marhomy,. Abdel-Sattar, Stability analyi of rotor-bearing ytem via Routh-Hurwitz riterion, Appl. Energ. 77(4) [] A. El-Marhomy, Parametri tability analyi of rotor-bearing ytem, Pro. atl. Si. Coun. ROC (A). 3(999b) [3] A. El-Marhomy, Dynami and tability of elati haft-bearing ytem with nonlinear bearing parameter, Heat. Ma. Tranfer. 3(999a) [4] T. Lim, R. Singh, Vibration tranmiion through rolling element bearing, part I: earing tiffne formulation, J. Sound. Vib. 39(99) [] J. Sinou, on-linear dynami and ontat of an unbalaned flexible rotor upported on ball bearing, Meh. Mah. Theory. 44(9) [6] J. Sinou, F. Thouverez, on-linear dynami of rotor-tator ytem with non-linear bearing learane, Cr. Meanique. 33( (4) [7] M. raniky, Stability of withed and hybrid ytem, Pro. 33rd IEEE Conf. Deiion Control, 4 (994) [8] M. raniky, Multiple lyapunov funtion and other analyi tool for withedand hybrid ytem, IEEE. T. Automat. Contr. 43(998) [9] J. Hepanha,, Uniform tability of withed linear ytem: extenion of LaSalle' invariane priniple, IEEE. T. Automat. Contr. 49(4) [] D. Liberzon, J. Hepanha, A. More, Stability of withed ytem: a Lie-algebrai ondition, Syt. Control. Lett. 37(999)7-. [] K. Gerriten, A. Van der Shaft, W. Heemel, On withed Hamiltonian ytem, Pro. MTS.() diana, U.S.A. [] L. Zhu, Y. Wang, Study on the tability of withed diipative Hamiltonian ytem, Si. China. Ser. F: form. Siene 49(6 ) ternational Sholarly and Sientifi Reearh & novation 6(3) 686