ANALYSIS OF ROTOR-BEARING SYSTEM USING THE TRANSFER MATRIX METHOD

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ANALYSIS OF ROTOR-BEARING SYSTEM USING THE TRANSFER MATRIX METHOD Mohammad T. Ahmadian Department of Mehanial Engineering Omid Ghasemaliadeh Department of Mehanial Engineering Center of Eellene in Dnamis, rootis, and Automation, Sharif Universit of Tehnolog, Tehran 11365-9567, Iran. Hossein Sadeghi Department of Mehanial Engineering Mohammad Bonadar Department of Mehanial Engineering ABSTRACT One of the methods to find the natural frequenies of rotating sstems is the appliation of the transfer matri method. In this method the rotor is modeled as several elements along the shaft whih have their own mass and moment of inertia. Using these elements, the entire ontinuous sstem is disretied and the orresponding differential equation an e stated in matri form. The earings at the end of the shaft are modeled as equivalent spring and dampers whih are applied as oundar onditions to the disretied sstem. In this paper the dnamis of a rotor-earing sstem is analed, onsidering the grosopi effet. The thiness of the dis and earings is also taen into aount. Continuous model is used for shaft. Results Show that, the stiffness of the shaft and the natural frequenies of the sstem inrease, while the amplitude of viration dereases as a onsequene of inreasing the thiness of the earing. 1 INTRODUCTION Rotating shafts have an important role in power transmission and are used etensivel in mahines suh as turines, generators, eletrial motors and et. Various methods have een used in analing rotar sstems. The Jeffott model [1] for rotors assumes a massless shaft having a rigid rotor on it. The transfer matri is a useful method for analing elasti strutures []. The transfer matries of man standard parts are given in handoos. Sometimes, however, the required transfer matri is not atalogued. It is often possile to find the transfer matri using simple dnami equations or using results that are taulated in engineering handoos. The merit of transfer matri method lies in the fat that the dnamis of ompliated sstems an e simplified into a sstem of algerai equations. Prohl [3] in 195 suggested the transfer matri method. lund and orutt [] in 1967 used this method with the ontinuous model for shaft and unalane in dis taen into aount. Kir and Bansal [5] in 1975 used the transfer matri method for alulating damped natural frequenies. Kang and Tan [6] in 1998 investigated disontinuit in shafts using this method. In this paper a Continuous model is used for the shaft and the grosopi effet is also taen into aount. The e feature of this stud is that the thiness of the dis is onsidered and the width of supporting end earings is also modeled. dnamis of the shaft. This theor assumes a onstant transversal shear strain at the diretion of thiness of the eam. To determine the shear fores, a orretion fator (K) is used. Eah setion of the shaft is assumed to have two translational and two rotational motions aout the perpendiular aes. In this paper the fluid film journal earing is onsidered for oth ends of the shaft. Sine the earings have muh more damping effet than the shaft, this effet is negleted for the shaft and the motion of shaft is assumed to e undamped. The earings are modeled as equivalent spring and dampers whih are applied as oundar onditions to the disretied sstem. The transfer matri of the shaft and the dis are otained assuming that the grosopi effet is present and the dis has a finite thiness. 3 SHAFT TRANSFER MATRIX An element of the defleted shaft in the and planes is depited in figs.1 and, respetivel. The susripts l and r orrespond to left and right sides of the element. The righthanded loal oordinate sstem 1, 1, 1 stis to the shaft having its 1 ais in the aial diretion of the shaft. As shown elow, the variation of shear fores and ending moments at the oth ends of the element is alulated using partial derivatives. Figure 1. Element of shaft in plane MODELING ASSUMPTIONS In this analsis the viration amplitude is assumed to e small. The Timosheno eam theor is used for epressing the 1

Figure. Element of shaft in plane The resultant moment vetors for the element with unit length ( d 1) is M Q M M Q (1) 0 and the angular momentum vetor is I t H I t () I Where and are the rotation of shaft aout and ais, repetivel and are assumed to e small as a onsequene of assuming the viration amplitude to e small, and is the spin veloit and I and I are the seond moments of area whih are assumed to e equal. Newton s seond law for rotation is epressed as; [7] dh M ( ) ΩH (3) dt where T Ω () t t is the angular veloit of the loal oordinate sstem. Sustituting equs. 1, and into equ. 3 results in M Q I I I t t t (5) M Q I I I t t t the resultant eternal fore vetor eerted on the element is Q Q F (6) 0 Appling the Newton s seond law to the element results in Q, Q A A (7) t t Equations 5 and 7 onstitute the partial differential equation of the shaft. The following equations govern the ending moment and shear fore in the shaft. [8] M EI, M EI (8) Q KAG, Q KAG B replaing equ. 8 into equs. 5 and 7 the governing differential equations of the shaft in and planes are otained, whih simplifies to: KG E t KEG t 3 3 A 0 3 EI t E t KG t (9) KG E t KEG t 3 3 A 0 3 EI t E t KG t The general solution to these PDEs is in the form of: ˆ ˆ os ts sin t (10) os ˆ sin ˆ t s t where ˆ is the natural frequen of the sstem. Sustituting equ. 10 into equ. 9 results in four ODEs whih require that the fators of Sin and Cos to e ero in order to have a nontrivial solution. The solution to the aove set of ODE is in the form of: ue, s ue s (11) ve, s vse B sustituting the assumed solution into the four equations that have een reated, a set of algerai equations is otained whih is not stated here for the sae of revit. In order to have a nontrivial solution the determinant of the oeffiient matri should vanish whih results in the following harateristi equation: 3 A KG I I E I I KG KGEI KGI (1) 3 A KG I I E I 0 I KG KGEI KGI

The aove equation has the following eight roots: 1, i, 3, i, whih result the following statements for the four unnowns us, u, vs, v us v, u vs, fori, 1 (13) us v, u vs, fori, 3 Using the otained roots, the displaement amplitudes ma e epressed as; 1osh 1sinh 13os sin 5osh 36sinh 37os 8sin s 9osh 110sinh 111os 1sin 13 osh 31 sinh 315 os 16 sin (1) 1osh 1sinh 13os sin 5osh 36sinh 37 os8sin s 9osh 110sinh 111os 1sin osh sinh os sin 13 3 1 3 15 16 All the i oeffiients in the aove equation are olleted into the vetor B. B i 16 1 (15) To otain the transfer matri of the shaft it is neessar to epress the state vetor in one side of the element in terms of the state vetor in the other side. The state vetor for the shaft is: S { s s s s (16) T M Ms M M s Q Qs Q Qs} where s and indies orrespond to the sin and os epressions as stated for and in equ. 10. A vetor of position derivatives is also defined as; W { s s s s (17) T s s s s } At two ends of the element, the aove vetor is equal to: W 0 A0B (18) W A B L L where A 0 and are given in appendi 1. B omitting the Bvetor in equs. 16 we onlude that; 1 WL AL A0 WL A W 0 (19) B epanding the omponents in the W vetor, the relation etween W and S ma e found. Q M 1,, ( A ) KAG EI KAG 1 Q ( ) ( Q I I ) EI KG KAG Q M 1 (0),, ( A ) KAG EI KAG 1 Q ( Q I I ) ( ) EI KG KAG B olleting the aove relations into matri form, the following relation is resulted W F S (1) where F is given in appendi 1. Using equs. 19 and 1 the state vetor in one side of the element ma e related to that of the other side as follows: 1 1 S F W F A W L L 0 1 F AFS TS 0 0 () So the transfer matri of the shaft is otained from. 1 1 T F A A F (3) 1616 L 0 DISK TRANSFER MATRIX The dis free od diagram is depited in Fig. 3. () M l (M ) l Q l Q r (Q ) r h (Q ) l M r (M ) r 0 (0 ) Figure 3. Free od diagram of dis Here it is assumed that the dis is nonhomogeneous suh that its enter of mass is loated a small distane r off the geometri enter. It is also assumed that r is suh small that its effet on the inertia tensor ma e negleted. So the inertia tensor of the dis is written as: Id 0 0 I 0 Id 0 () 0 0 I d where I d is the mass moment of inertia of the dis aout an ais passing through its enter of gravit (CG). Again the Newton s seond law is used to otain the equations of motion for the dis. Appliation of Newton s seond law to translational motion of the dis results in: l r m( r os t) Ql Qr (5) l r m( r sin t) Ql Qr Newton s seond law in rotational motion is given in equ. 3. The angular momentum of dis is H I Ω (6) where Ω is the same as than in equ.. The otained angular momentum should e transformed into the gloal oordinate sstem using a rotation matri R t. M R( t) H Ω( R( t) H) (7) The resultant of moment vetors ating on the dis is:

h M Mr ( Q Qr) h M M r M ( Q Qr ) (8) 0 Sustituting equs. 6 and 8 into equ. 7 ields the equation of motion for rotation of the dis. Again writing the degrees of freedom as in equ. 10, the aove governing differential equations are simplified to some algerai equations. Using the following additional equations; r l, r l (9) r l h l, r l h l the state vetor at oth ends of the dis ma now e related and the transfer matri is otained. 5 BEARING TRANSFER MATRIX A linear model is assumed for earings whih uses two equivalent pairs of spring and dampers for eah earing as shown in fig.. X (Y) Q l M l t M r Q r K/ C/ K/ C/ Figure. Bearing model In this figure t represents the length of the shaft that lies in the aring and or is the foundation displaement. This model assumes that eah earing an withstand moment in addition to transversal load. In a simplified model we assume that the earing fore vetor an e written as; F f(, ) iˆg(, ) ˆj (30) where f (, ) (31) g(, ) the fore and moment resultants are as follows; 0 Z F Ql Qr l l r r l l r r F Ql Qr l l r r l l r r M M r M l l t t Ql Qr l t r t (3) r t l t l t r t r t M Ml Mr l t t Ql Qr l t r t r t l t l t r t r t Using the Newton s seond law in rotation and translation and using the following geometrial additional equations; r l, r l (33) r l t, r l t the governing differential equation of the earing is otained. Again assuming harmoni motion for all degrees of freedom (as equ. 10), the governing algerai equation is otained whih is used to alulate the transfer matri of the earing. To otain the transfer matri etween two desired points of the sstem, the transfer matries of the individual elements loated etween the points should e multiplied onsequentl. If the transfer matri of the whole sstem is availale, appling the oundar onditions on the state vetor of oth ends will result in a relation among the unnown properties of the sstem. 6 RESULTS To investigate the appliailit of this method in dnami analsis of rotor-earing sstems, an eample is provided. A shaft arring several diss is supported on two earings on oth ends as shown in fig. 5.

Figure 5. Shemati figure of a rotor In this eample a fleile shaft is arring three similar diss and two earings are supporting the shaft. The parameters of the sstem are given in tale 1. Tale 1. Parameters of the rotor-earing sstem. Modulus of elastiit N / m 7.07 10 Shear modulus N / m 7 1.9 10 3 Densit g / m 3 7.75 10 Dis mass g 13.7 Polar moment of inertia g. m 100 Mass moment of inertia g. m 51 Unalane of dis g. m 0.0137 Shaft diameter m When the dis width is ignored the amplitude-rotational speed urve in fig. 6 is otained. Figure 6. response of the sstem when onsidering the dis width It is shown that the first natural frequen of the sstem has inreased ompared to the previous ase. The inrease of the natural frequen is attriuted to the fat that onsidering the dis width, the effetive length of the shaft dereases and hene the sstem eomes more rigid. The derease in the amplitude of viration is another onsequene of thiening the dis. The real ase an e ahieved when oth the dis and earing thinesses are taen into aount. Figure 7 orresponds to this ase. Figure 5. response of the sstem when ignoring the dis width In the aove figure the pi amplitude orresponds to the first natural frequen of the sstem whih agrees with approimate methods availale suh as the Dunerle s method. When the thiness of the dis is also taen into aount, the response urve hanges signifiantl as depited in fig. 6. Figure 7. response of the sstem when onsidering oth the dis and earing widths It is oserved from fig. 7 that onsidering the width of earing the natural frequen inreases again while the amplitude of viration has dereased. 7 CONCLUSIONS In this paper the matri transfer method is applied to rotorearing sstem. The transfer matri of individual elements inluding the shaft, dis and the earing are otained. The

merit of this approah over the onventional approimate methods is that the thiness of the diss and the earings ma also e onsidered. It is shown that the thiness of diss and earing have a onsiderale effet on the viration ehavior of the sstem. This method an e applied to rotor-earing sstems with various onfigurations. 6 REFERENCES [1] Rao, S.S., 1983, Rotor Dnamis, John Wile & Sons, Newor [] Pestel, E. C., and Leie, F. A., 1963, Matri Methods in Elastomehanis, MGrawHill, New or. [3] Prohl, M. A., 195, "A General Method for Calulating Critial Speeds of Fleile Rotors", ASME J. Appl. Meh., Vol. 67, p. 1. [] Lund, J. W., and Orut, F.K., 1967, Calulations and Eperiments on the Unalane Response of a Fleile Rotor, ASME Journal of Engineering for Industr, Vol. 89 (), pp.785-596. [5] Bansal, P. N., and Kir, R. G., 1975, "Stailit and Damped Critial Speeds of a Fleile Rotor in Fluid Film Bearings," ASME J. Ind., Series B, 97(), pp. 135 133. [6] Tan, C.A., Kang. B., 1999, Free viration of aiall loaded, rotating Timosheno shaft sstems the wavetrain losure priniple, International Journal of Solids and Strutures, Vol 36, pp. 031-09. [7] Meriam, J.L, Kraige L.G, 1998, Dnamis, John Wile & Sons, New or. [8] Beer, F.P, Johnston, J.R., 199, Mehanis of materials, M Graw Hill, Singapore. APPENDIX 1- SOME TRANSFER MATRICES 1 0 1 0 1 0 1 0 0 1 0 0 3 0 1 0 0 3 0 0 3 3 3 3 0 1 0 0 3 0 1 0 1 0 1 0 1 0 0 1 0 0 3 0 1 0 0 3 0 0 3 3 3 3 0 1 0 0 3 0 A0 1 0 1 0 1 0 1 0 0 1 0 0 3 0 1 0 0 3 0 0 3 3 3 3 0 1 0 0 3 0 1 0 1 0 1 0 1 0 0 1 0 0 3 0 1 0 0 3 0 0 3 3 3 3 0 1 0 0 3 0 1...8,1...8 osh( 1L) sinh( 1L) os( L) sin( 1L) osh( 3L) sinh( 1L) os( L) sin( 1L) sinh( 1L) 1 osh( 1L) 1 sin( L) os( L) sinh( 3L) 3 osh( 3L) 3 sin( L) os( L) osh( 1L) 1 sinh( 1L) 1 os( L) sin( L) osh( 3L) 3 sinh( 3L) 3 os( L) sin( L) 3 3 3 3 3 3 3 3 sinh( 1L) 1 osh( 1L) 1 sinh( L) os( L) sinh( 3L) 3 osh( 3L) 3 sinh( L) os( L)

1...8,9...16 osh( L) sinh( L) os( L) sin( L) osh( L) sinh( L) os( L) sin( L) 1 1 3 3 sinh( 1L) 1 osh( 1L) 1 sin( L) os( L) sinh( 3L) 3 osh( 3L) 3 sin( L) os( L) osh( 1L) 1 sinh( 1L) 1 os( L) sin( L) osh( 3L) 3 sinh( 3L) 3 os( L) sin( L) 3 3 3 3 3 3 3 3 sinh( 1L) 1 osh( 1L) 1 sin( L) os( L) sinh( 3L) 3 osh( 3L) 3 sin( L) os( L) 9...16,1...8 osh( 1L) sinh( 1L) os( L) sin( L) osh( 3L) sinh( 3L) os( L) sin( L) sinh( 1L) 1 osh( 1L) 1 sin( L) os( L) sinh( 3L) osh( 3L) sin( L) os( L) osh( 1L) 1 sinh( 1L) 1 os( L) sin( L) osh( 3L) 3 sinh( 3L) 3 os( L) sin( L) 3 3 3 3 3 3 3 3 sinh( 1L) 1 osh( 1L) 1 sin( L) os( L) sinh( 3L) 3 osh( 3L) 3 sin( L) os( L) 9...16,9...16 osh( 1L) sinh( 1L) os( L) sin( 1L) osh( 3L) sinh( 1L) os( L) sin( 1L) sinh( 1L) 1 osh( 1L) 1 sin( L) os( L) sinh( 3L) 3 osh( 3L) 3 sin( L) os( L) osh( 1L) 1 sinh( 1L) 1 os( L) sin( L) osh( 3L) 3 sinh( 3L) 3 os( L) sin( L) 3 3 3 3 3 3 3 3 sinh( 1L) 1 osh( 1L) 1 sin( L) os( L) sinh( 3L) 3 osh( 3L) 3 sin( L) os( L)

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