CHAPTER 8 ROTORS MOUNTED ON FLEXIBLE BEARINGS

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1 CHAPTER 8 ROTORS MOUNTED ON FLEXIBLE BEARINGS Bearings commonly sed in heavy rotating machine play a significant role in the dynamic ehavior of rotors. Of particlar interest are the hydrodynamic earings, which are self acting and capale of carrying heavy loads. Becase the thin film that separates the moving srfaces spports the rotor load, these earing acts lie a spring and provide damping de to sqeee film effect. The stiffness and damping properties of the oil film significantly alter the critical speeds and ot-of-alance-response of a rotor, lowering its critical speed y p to 5 % in practice. In addition, rotor instaility occrs, which is self excited viration arising ot of the earing flid film effects and this is an important factor to e considered in the rotor design. In flid film earings, the hydrodynamic pressre is generated etween the earing and jornal srfaces y the motion of the jornal, from which the resltant dynamic earing force can e derived. For small viration aot the steady state eqilirim position of the jornal, we can define linearied earing stiffness and damping coefficients. In this chapter, the flid film earing characteristics (coefficients) are discssed for short earing approximation. Natral whirl freqency and staility analyses are done for speed independent and speed dependent earing characteristics. Gyan condensation scheme is tilied to decrease the sie of system matrices in case of forteen-element model. 8. Flid Film Bearing Characterstics for Short Bearing Approximation Flid film earing stiffness & damping coefficients, direct as well as cross-copled, stiffness and damping can e derived ased on short earing approximation (where pressre variation in the circmferential direction is assmed to e negligile compared with that in the axial direction and converse applies for long earing approximation; Lee, 993). The stiffness and damping coefficients are fnctions of the eccentricity (load), the rotational speed and the temperatre. If we evalate the linearied coefficients, assming small changes in displacements and velocities from a steady state eqilirim position as shown in Fig. 8., we may write the eqilirim eqation as f y v c c v f w c c w (8.) where ij and c ij are the stiffness and damping coefficients, with i representing the direction of force and j representing the direction of displacement and velocity. The eight linearied stiffness and

2 damping coefficients depend on the steady state operating conditions of the jornal, and in particlar, pon the rotational speed. Figre 8. Locs of jornal eqilirim position for short earing (Lee, 993) For the short earing, the dimensionless earing stiffness and damping coefficients as a fnction of the steady state eccentricity ratio of earing, ij ij C/W, C ij c ij Cω/W, i, j y,, are given y Lee (993), 4 { π + ( 3+ π ) ε + ( 6 π ) ε } ( ε ) 4 Q( ε ) ; 4 4 { π + ( 3+ π ) ε + ( 6 π ) ε } ε ( ε ) Q( ε ) 4 π { π π ε ( 6 π ) ε } Q( ε ) ; 4{ π ( 6 π ) ε } Q( ε ) ε ( ε ) + and (8.) 4 π{ π + ( 4 π ) ε + π ε } Q( ε) C ; C C 8{ π + ( π 8) ε } Q( ε ) ε ( ε ) C where ( ε ){ π + ( π 8) ε } π Q( ε ) (8.3) ε µ N r µ DLN r Q ( ε ) ; S. P C W C { π ( ε ) + 6ε } 3 / 36

3 To determine the stiffness and earing coefficients of a short earing, we first determine the Sommerfield nmer, S. Here W is the earing radial load, D is the jornal diameter, L is the length of earing, µ is the viscosity of lricant at operating temperatre (sally given in the nit of centipoise, cp.0054(0) -3 Nsec/m ), ω πn the rotational speed of jornal, N the nmer of revoltions per second, and C is the radial clearance. The eccentricity of the jornal center defined as, ε e/c, where e is the jornal eqilirim eccentricity. We can then determine the eccentricity ratio nder steady state operating conditions y ( ) ( ) L ε S D πε π ε + 6ε (8.4) While determining the eccentricity ratio, an iteration procedre is reqired, as eqation (8.4) is transcendental in natre. A plot of S as a fnction of eccentricity for L/D0.5 short earing according to eqation (8.4) is given in Fig. 8.. Fig 8. Sommerfield nmer of short earing as a fnction eccentricity ratio From the given geometric parameters of earing and its operating conditions the Sommerfield nmer, S, can e otained. From Figre 8. eccentricity ratio, ε, can e otained for a particlar Sommerfield nmer. Once the eccentricity ratio is otained the stiffness and damping coefficients are otained from eqations (8.) and (8.3). These are given (For L/D 0.5, earing load, W 8.88 N, radial clearance, C cm, and viscosity at operating temperatre, µ 0.04 N sec/m ) as a fnction of speed in Tale 8. & in Figre 8.3 for direct and cross-copled stiffness coefficients and in Tale 8. & in Figre 8.4 for direct and cross-copled damping coefficients (these vales will e sed for illstration in section 8.4). 36

4 Tale 8. Speed in Stiffness coefficients variation with respect to rotational speed Stiffness coefficients N/m (Direct and cross-copled) rpm e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+07 Tale 8. Speed in Damping coefficients variation with respect to rotational speed Damping coefficients N sec/m (Direct and cross-copled) rpm C C C C e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+04 Fig 8.3 Stiffness coefficients variation with respect to rotational speed Fig. 8.4 Damping coefficients variation with respect to rotational speed 363

5 8. FEM Formlation for Bearings The classical linearied model with eight spring and damping coefficients is employed for the modelling of earings in the present wor. In this model, the forces at each earing are assmed to oey the governing eqations of the following form c c { q } { q } { F } c c + (8.5) where { } { } T q v w is the earing displacement vector and c ij and ij are the earing and damping coefficients, and {F } is the vector of earing forces. 8.3 Natral Whirl Freqency and Staility Analysis The earing eqations of motion (8.5) inclded in the eqations of motion of the rotor sstrctre, which inclde the rotary inertia, shear and gyroscopic effects. The resltant system eqations of motion then ecomes s s s s s s [ M ]{ q } + ([ C ] [ G ]){ q } + ([ ] + [ ]){ q } { 0} ω (8.6) where [C ] and [ ] are the earing damping and stiffness matrices respectively. To determine the whirl speeds and staility eqation (8.6) can e rewritten as where [ ]{ h} + [ B]{ h} { 0} A (8.7) [ A] [ ] [ ] s s 0 M M 0, [ B] s s s M C ωg [ 0] and { h} { } { } q q (8.8) The associated eigen vale prolem for eqation (8.7) is soght from an assmed soltion form as { } { } h h e λ (8.9) 0 t On sstitting eqation (8.9) into eqation (8.7), we get [ 0] [ I] + { } { [ ] } { } { } s s s s h s 0 h0 M C ω G λ (8.0) 364

6 where λ α + j p is the complex eigen vales and p is the natral whirl freqency. The real part of the eigen vale α indicates the damping in the system at the given speed and it is associated with each whirl speed. The parameter of logarithmic decrement, δ, is defined as πα δ (8.) p where δ represents the instaility threshold when δ < 0. The response of a dynamic system is a decay fnction, which involves the damping term. In order to get stale response the amplitde of viration shold decay as time increases. This will happen if the damping index (α < 0) is negative. Eigen vales are of the form v α ± ip where p is the natral whirl freqency (speed dependent) and α is the damping. The logarithmic decrement δ is defined as πα δ p where α is negative stale δ is positive α is positive nstale δ is negative δ > 0 stale δ < 0 nstale Campell diagram can e drawn similar to gyroscopic cople effects. Here also forward and acward whirls will occr. Staility can e checed y oserving the sine of logarithmic decrement. Eigen vectors will e complex. Example 8. Otain the assemled system eqations of motion y the finite element method for transverse virations of rotor-earing system as shown in Figre 8.5. Consider shaft as continos system i.e. mass and stiffness is distrited continosly throghot the shaft. The shaft is of m of span and the diameter is 0.05 m with the mass density of 7800 g/m 3. The shaft is spported at ends y flexile earings as shown in Figre 8.5. Consider the motion in oth the vertical and horiontal planes. Discretise the shaft into one-element and show elemental eqations for the shaft and earings. Tae the following earing properties: 365

7 366 For earing A: xx 0 MN/m, 5 MN/m, xy -.5 MN/m, yx 5 MN/m, c xx 00 N-s/m, c xy 50 N-s/m, c yx 40 N-s/m, c 400 N-s/m, and for earing B: xx 4 MN/m, 7 MN/m, xy -.5 MN/m, yx 30 MN/m, c xx 0 N-s/m, c xy 60 N-s/m, c yx 35 N-s/m, c 380 N-s/m. shaft earings A B Figre 8.5 A rotor monted on flexile spports Otained the Campell diagram to show critical speeds and staility thresholds. Soltion: EOM of the earing at location in Figre 8.5() can e written as + y y y x f f c c c c (A) (a) A rotor-earing system () Beam element in two orthogonal planes Figre 8.5 A niform shaft monted on flexile spports EOM of the earing at location can e written as + y y y x f f c c c c (B) x y x y x

8 Eqations (A) and (B), can e expanded as and c 0 c 0 y 0 0 y f y θ θ 0 + c 0 c f θ y θ y 0 c 0 c 0 y 0 0 y f y θ θ 0 + c 0 c f θ y θ y 0 (C) (D) Comining eqations (C) and (D), we get the EOM of earings, as c 0 c y f y θ f θ y y f y θ f θ y θ y 0 y θ c 0 c θ y c 0 c 0 + y θ c 0 c 0 (E) EOM of the shaft element considering the single element as shown in Figre 8.5(), is given as y θ ρ AL θ y y θ 56 sym 4 θ y 367

9 S θ M S EI θ M + 3 l S θ M S θ M y y y y y y y y (F) Comining eqations (E) and (F), we get eqations of motion of the rotor-earing system, as y c 0 c y θ θ c 0 c ρ AL θ y θ y y c 0 c 0 y θ θ c 0 c 0 4 θ θ y y ( ) y S y + f y θ M 0 0 ( + ) S + f 0 EI θ y M y l ( + ) 6 / 0 + y S y + f y 0 θ M ( + ) 6 S + f 0 θ y M y (G) In the right hand side the first vector represents reaction forces and moments and second vector is corresponding to the external forces and moments. The reaction force vector will ecome ero. [ M]{ } + [ c]{ } + [ ]{ } { 0} (H) The standard form of the eigen vale prolem is { h } [ D]{ h} (I) 368

10 where [ D] 0 [ ] M [ c] [ M] [ ] ; { h} { } { } (J) Let { } { h } e vt h 0 () Eqation (I) ecomes { } [ ]{ } v h D h (L) Nmerical Examples and Discssions To demonstrate the application of finite element model, a typical rotor earing system as illstrated in Fig. 4. is analed to determine the whirl speeds and staility. The distrited rotor is modelled as seven and forteen element memer and oth the earings are to e taen identical. The geometric and physical data of these elements are same as of Example 4.4. The flid film earings are idealied as a linear short earing (L/D 0.5) and are located at stations two and seven for the case of sevenelement model and three and thirteen for the case of forteen-element model. The following two cases of earing characteristics are analed: (a) Speed independent earing characteristics and () Speed dependent earing characteristics. (a) Speed independent earing characteristics. Bearing geometric and physical data are as follows: diameter of the earing, D.54 cm, length of earing, L.7 cm, (i.e. L/D 0.5), radial clearance, C cm, and viscosity at operating temperatre, µ 0.04 N sec/m. The direct as well as the cross-copled stiffness and damping coefficients are fond at spin speed 4000 rpm. These are 369

11 N/m, N/m, N/m, N/m C C C C N sec/m, N sec/m, N sec/m These earing characteristics are taen as constants for all other speeds of the rotor i.e. these vales are treated as speed independent characteristics of the earing for illstration (t we cold have tae earing characteristics at any other spin speeds). In order to illstrate a Jeffcott rotor model (simply spported massless shaft with a central dis) monted on flexile identical earings having direct and cross-copled stiffness coefficients (withot damping) is analed. Critical speeds from the classical closed form soltions (Rao, 996; Appendix 8.) and from the present analysis are talated in Tale 8.3 for the comparison. It shows that the present code reslts are in excellent agreement with the classical soltions. Tale 8.3 Comparisons of critical speeds from classical soltions and from the present analysis for Jeffcott rotor model (simply spported massless shaft with a central dis) monted on flexile identical earings Critical speeds from the classical soltions* (rad/sec) Critical speeds from the present FEM code (rad/sec) p (7)# * (Appendix 8.), # Vales in parentheses denote the nmer of elements considered. The whirl freqencies a fnction of spin speed and the corresponding logarithmic decrements, for the rotor system shown in Figre.3, are talated in Tale 8.4 for seven-element rotor model. The whirl freqency map and staility are presented in Fig The letters F and B refers to forward and acward whirl modes, respectively. The first for critical speeds in forward & acward direction and logarithmic decrement are talated in Tale 8.5 for seven and forteen element models. It is oserved that third & forth forward whirl mode and all the acward whirl modes are stale for entire spin-speed envelope stdied. Condensation scheme is performed in the case of forteenelement rotor model. Here all the rotational degrees of freedoms except at lmped mass and earing stations taen as slave degrees of freedom (total no. stations 5,, no. of DOF at each node 4, total no. of lmped stations 4 (at stations 5, 7, 9 & ), total no. of earing stations (at stations 3 and 3), no. of rotational DOF at each node, total no. slave degrees of freedom (5-6) 8, final DOF (5 4) (9 ) 4, final order of eigen vale prolem 4 84). The first for critical speeds in forward & acward direction and logarithmic decrement are also talated in Tale 8.5. The reslts show that p to second mode agreement is excellent as compared to the withot condensation scheme, and for third & forth mode agreement is qite good. The comparison of the 370

12 critical speeds when the rotor is monted on flid film earings is made with the critical speeds when the rotor is monted on rigid earings and is given in Tale 8.6. The comparison shows that with the earing characteristics the natral whirl freqencies are decreased especially for higher modes. Tale 8.4 Natral whirl freqency and logarithmic decrement as fnction of spin speed of a rotorearing system spported on speed independent earings. Spin speed (rad/sec) Natral whirl freqencies (rad/sec) Logarithmic decrement δ Forward Bacward Forward Bacward Fig. 8.5 Natral whirl freqency map and staility of rotor earing system spported on speed independent earings. 37

13 Tale 8.5 Critical speeds & logarithmic decrement of a rotor system spported on speed independent earings Mode Critical speeds (rad/sec) and (Logarithmic decrement δ) No. P (7) p (4) #p (4) Forward Bacward Forward Bacward Forward Bacward 5.00 ( )* 7.8 (0.0057) 5.0 ( ) 7.9 (0.0057) 5.79 (0.0060) 7.48 ( ) ( ) 47.0 (0.05) ( ) (0.05) ( ) 45.8 (0.04) (0.339) (0.3347) ( ) (0.3337) (0.339) ( ) (0.0806) (0.0058) ( ) ( ) 90.5 ( (0.0039) * The vales in the parenthesis are logarithmic decrements, # Vales otained from condensation scheme y taing the 8 rotational degrees of freedom as slaves at all nodes except at lmped mass earing nodes. Tale 8.6 Comparison of critical speeds for the rotors monted on rigid earings against the flid film earings. Critical speeds (rad/sec) Critical speeds (rad/sec) % decrease in Mode with rigid earings with flid film earings critical speeds No. Forward Bacward Forward Bacward (for forward whirl) (a) Speed dependent earing characteristics: The direct as well as cross-copled stiffness and damping coefficients are considered as speed dependent. These are calclated from the eqations (8.) and (8.3) as a fnction of speed and given in Tale 8. and 8. (For L/D 0.5, earing load, W 888. N, radial clearance, C cm, and viscosity at operating temperatre, µ 0.04 N sec/m ). The natral whirl freqencies, as a fnction of spin speed is determined from the eqation (8.0) for rotor-earing system as shown in Figre.3. These are talated along with the logarithmic decrement at each spin speed in Tale 8.7 for seven-element model and are shown in Fig 8.6. The first for critical speeds and corresponding logarithmic decrement are talated in Tale 8.8 for seven and forteen element models. It is oserved that third & forth forward whirl mode and all the acward whirl except forth modes are stale for entire spin-speed envelope stdied. Condensation scheme is performed in case of forteen-element model. The selection of slaves and order of eigen vale prolem is same as the case (a). Reslts show that p to second mode agreement is excellent as compared to the withot condensation scheme, and for third & forth mode agreement is qite good. At the lower speeds (elow 000 rpm) the natral whirl freqencies happens to e infinite. We wold not ale to explain this ehavior. 37

14 Fig 8.6 Whirl freqency map of rotor earing system spported on speed dependent flid film earings. Tale 8.7 Natral whirl freqencies and logarithmic decrement as fnction of spin speed of a rotor earing system spported on speed dependent flid film earings. Spin speed (rad/sec) Natral whirl freqencies (rad/sec) Logarithmic decrement δ Forward Bacward Forward Bacward * * * Natral whirl freqency ecomes infinity Tale 8.8 Critical speeds & logarithmic decrement of a rotor system spported on speed dependent earings Mode Critical speeds (rad/sec) and ( Logarithmic decrement δ) No. p (7) p (4) p (4 Forward Bacward Forward Bacward Forward Bacward

15 (0.048) (0.003) (0.03) (0.003) (0.04) (0.004) ( ) (0.049) ( ) (0.04) ( ) 46.5 ( ) (0.574) (0.387) (0.9) (0.304) (0.49) 69.6 (0.377) (0.0778) ( ) (0.070) 87. (-0.007) (0.0778) 87.5 ( ) APPENDIX 8.: Rotor monted on flexile earings For a simple Jeffcott rotor monted on flexile identical earings (having direct and cross-copled stiffness coefficients with no damping) the freqency eqation can e written as (Rao, 996): where and p, ω + ω ± ω ω + µ µω ω ω ; ω ; µ ; µ m m [ ( + ) 4 ] ( + )( + ) 4 [ ( + ) 4 ] ( + )( + ) 4 ( + )( + ) 4 ( + )( + ) 4 Here the following parameters are chosen:. 0 9 N/m & N/m are the direct stiffness coefficients, respectively and N/m & N/m are the cross-copled stiffness coefficients, respectively. These are otained for short earing (L/D 0.5) at spin speed ω 4000 rpm (these are same as those for speed independent earing). And (ω n ) m is the stiffness of the simple Jeffcott rotor, ω n 45.7 rad/sec (chosen from Tale. for first mode) is the rigid earing critical speed and m 4. g (total mass of for diss) is mass of the central dis monted on the shaft. The length and the diameter of the shaft are 3.5 m and m, respectively. REFERENCES Lee C.-W., 993, Viration Analysis of Rotors, lwer Academic Plishers, London. Rao, J. S., Rotor Dynamics, New Age International Plishers, Third Edition, 996. Exercise 8. A long rigid symmetric rotor is spported at ends y two identical earings. Let the shaft has the diameter of 0. m, the length of shaft is m and the mass density of the shaft material 374

16 eqal to 7800 g/m 3. The earing dynamic characteristics are as follows: xx N/mm with other stiffness and damping terms eqal to ero. By considering the gyroscopic effect also, otain whirl natral freqencies of the system, if rotor is rotating at 0, 000 rpm. Exercise 8. Find the critical speeds of the rotor earing system shown in Figre E8.. The shaft is made of steel with Yong s modls E. (0) N/m and niform diameter d 0 mm. Treat the shaft as flexile and massless. The mass of the disc is: m d g with negligile diamentral mass moment of inertia. Bearings B and B are identical earings and having the following properties:. N/m,.8 N/m, 0. N/m and 0. N/m. The varios shaft lengths are as follows: B D 75 mm, and DB 50 mm. Give the detailed steps involved in formlation of the system eqations. Y X Z B D B Figre E8. Exercise 8.3 Otain critical speeds of a rotor-earing system as shown in Figre E8.. The shaft is made of steel with Yong s modls E. (0) N/m, mass density of ρ 7800 g/m 3 and niform diameter d 0 mm. Treat the shaft as flexile and having distrited mass. The mass of the disc is: m d g with the diamentral mass moment of inertia I d 0.0 g-m and negligile polar mass moment of inertia (i.e. gyroscopic effects neglected). Bearings B and B are identical earings and can e assmed as short earing with L/D 0.5, D 0 mm and c r µm. The inetic 0 viscosity, µ, of the lricant is 8 centi-stoes at operating temperatre ( 40 C ) and the specific gravity is Plot and se the short earing dynamic parameters as given in Appendix 8.. The varios shaft lengths are as follows: B D 75 mm, and DB 50 mm. Give the detailed steps involved in formlation of system eqations. Exercise 8.4 (a) Find the ending critical speed of the system shown in Figre E8. in which the disc is made of solid steel with a diameter of 7 mm, and a thicness of 5.4 mm. The mass density of the steel is 7800 g/m 3. The disc D is placed in the middle of a shaft and the total length of the steel shaft etween earings is 508 mm, and its diameter is.7 mm. The earings have eqal flexiility in all directions, the constant for either one of them eing 75 N/m. () Solve the same prolem as 375

17 part (a) except that the earings have different vertical and horiontal flexiilities: ho 75 N/m and ho 350 N/m for each of the earings. Neglect the cross-copled stiffness coefficients and the mass of the shaft. Exercise 8.5 Draw the first three modes, associated with the lowest three natral freqencies of a flexile niform shaft as hwon In Figre E8.5, with increasing spport stiffness i.e. (i) 0 (ii) 0 < < (iii). Figre E