Analysis of whirl speeds for rotor-bearing systems supported on fluid film bearings

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Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 18 (2004) 1369–1380 www.elsevier.com/locate/jnlabr/ymssp

Analysis of whirl speeds for rotor-bearing systems supported on fluid film bearings Madhumita Kalitaa, S.K. Kakotyb,* b

a Assam Engineering Institute, Guwahati 781003, India Department of Mechanical Engineering, Indian Institute of Technology, North Guwahati 781 039, India

Abstract There are several numerical approximations for vibration analysis of rotor bearing systems and the most popular approach is the Finite Element Method. In the light of the above it is proposed to undertake a study on the dynamic behaviour of Timoshenko beam supported on hydrodynamic bearings incorporating internal damping using FEM model. Critical speeds are estimated for synchronous whirl at different operating conditions using Campbell diagrams. It is observed in the analysis that in addition to the natural whirl frequencies, for every spin speed another whirling frequency appears in the solution, which is around half the spin speed. In case of fluid film bearings, half whirl is very common phenomenon. In case of dynamic coefficients evaluated using short bearing approximation, it is observed that these additional frequencies are of the same order as that of the synchronous whirling frequencies. The additional frequencies around half the spin speeds are found using only finite bearing dynamic coefficients; this clearly indicates the deficiency in the use of short bearing approximation in similar work. r 2003 Published by Elsevier Ltd.

1. Introduction Rotating machines are extensively used in diverse engineering applications. The accurate prediction of dynamic characteristics is important in the design of any type of rotating machinery. There have been many studies relating to the field of rotor dynamics during the past years. Out of the published works, the most extensive portion of literature on rotor dynamics is concerned with determination of critical speeds, natural whirl frequencies, instability thresholds and imbalance response. Several numerical approximations have been successfully developed to analyse the dynamic behaviour of rotor systems. However, the most popular approach well suited for modelling large-scale and complicated rotor systems is the Finite Element Method. Ruhl [1] *Corresponding author. E-mail addresses: madhumita [email protected] (M. Kalita), [email protected], [email protected] (S.K. Kakoty). 0888-3270/$ - see front matter r 2003 Published by Elsevier Ltd. doi:10.1016/j.ymssp.2003.09.002

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and Ruhl and Booker [2] are the first to utilise the finite element method to study the stability and unbalance response of turbo rotor systems. In their analyses, only elastic bending energy and translational kinetic energy were included. Many effects, such as the rotary inertia, gyroscopic moments, shear deformations, internal and external damping were neglected. Thorkildsen [3] included rotary inertia and gyroscopic moments in the finite element model. Nelson and McVaugh [4] generalised Ruhl’s work by utilising a finite element formulation including the effects of rotary inertia, gyroscopic moments and axial load to model a flexible rotor system supported on linear stiffness and viscous damping bearings. The work of Zorzi and Nelson [5] was the generalisation of the work of Nelson and McVaugh [4] by including both internal viscous and hysteretic damping in the same finite element model. Nelson [6] utilised Timoshenko beam theory for establishing shape functions and, thereby included transverse shear effects. Ozguven and Ozkan [7] presented the combined effects of shear deformation and internal damping to analyse the natural whirl speeds and unbalance response of rotor-bearing system. Chen and Ku [8] developed a C0 class Timoshenko beam finite element model and Ku [9] included internal damping in the same model. Rao [10] provided analysis of dual rotor supported on fluid film bearings. In view of the above the present investigation is made to find out the behaviour of rotor-bearing system supported on fluid film bearings. A finite element formulation is carried out [10] for Timoshenko beam incorporating translational inertia, rotary inertia, bending deformation, shear deformation, gyroscopic effect and internal damping. Stiffness and damping coefficients of the bearings are estimated based on formulation of Lund [11].

2. Theory 2.1. Finite element formulation of rotors A rotor-bearing system is composed of a uniform shaft of length l rotating at a constant speed O (Fig. 1) and supported by two bearings. It is assumed that as compared to translational motion, axial motion is negligible. Fixed reference is X–Y–Z and rotating reference is x–y–z. A typical cross-section of the shaft, in a deformed state, located at a distance x from the left end can be described by translations V ðx; tÞ; W ðx; tÞ and rotations Bðx; tÞ; Gðx; tÞ in Y and Z directions. The relationships can be expressed as V ðx; tÞ ¼ Vb ðx; tÞ þ Vs ðx; tÞ; W ðx; tÞ ¼ Wb ðx; tÞ þ Ws ðx; tÞ; Bðx; tÞ ¼ qWb ðx; tÞ=qx; Gðx; tÞ ¼ qVb ðx; tÞ=qx;

ð1Þ

where Vb; Vs ; and Wb ; Ws are translations due to bending and shear in the Y and Z directions, respectively. Timoshenko beam includes all the effects, i.e., bending deformation, rotary inertia, gyroscopic effect and shear deformation. The strain energy is due to bending and shear. Translation, rotation and gyroscopic effect contribute towards kinetic energy. In the present model each element has

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Fig. 1. Displacement variables and coordinate system.

two nodes and each node has four generalised displacements. The nodal displacement vector is given by fqge ¼ fV1 ; W1 ; B1 ; G1 ; V2 ; W2 ; B2 ; G2 gT :

ð2Þ

For shaft element, strain energy and dissipation function are given as in [9]. Shear correction factor k is given by k ¼ 6ð1 þ nÞ=ð7 þ 6nÞ with n as the Poisson’s ratio. Internal damping is included using the formulations of Ozguven and Ozkan [7], " # Za Zb ; ½Z ¼ Zb Za 1 þ ZH Za ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ Z2H ZH þ OZv ; Za ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Z2H

ð3Þ

ZH and Zv denote the viscous damping coefficient and hysteretic loss factor of the shaft material. Following Lagrangian approach, the elemental equation of motion is given by [9] ð½MT e þ ½MR e Þfqg . e þ ðZv ½K e  O½G e Þfqg ’ þ ðZa ½K e þ Zb ½KC e Þfqge ¼ fF e g; where ½K e ¼ ½Kb e þ ½Ks e ; ½G e ¼ ½H e  ½H eT ; Z l 0 e ½Kb ¼ ½Nb0 T EI½Nb0 dx; 0

ð4Þ

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½Ks0 e

¼

Z

l 0

½MT e ¼ e

½Ns0 T kGA½Ns dx;

Z Z

l

½Nt T rA½Nt dx;

0 l

½Nb T Id ½Nb dx; " # Z l e T 0 0 ½Nb dx: ½H ¼ Ip ½Nb 1 0 0

½MR ¼

0

ð5Þ

The equation of motion for disk is given as [9] ½M d fq. d g þ O½G d fq’ d g ¼ ½0 ;

ð6Þ

where ½M d and ½G d are mass and gyroscopic matrices of disk only. 2.2. Finite element formulation for bearings Eight spring and damping coefficients are employed for the modelling of bearings in the present work [9]. In this model, the forces at each bearing are assumed to obey the governing equations of the following form " # " # k k cyy cyz yy yz fq’ b g þ fqb g ¼ fF b g; ð7Þ czy czz kzy kzz where fqb g ¼ f v w gT is the bearing displacement vector and cij and kij are the bearing and damping coefficients; fF b g is the vector of bearing forces. The resultant system equation of motion then becomes ½M s fqg . e  ð½cb  O½G s Þfqg ’ þ ð½K s þ ½kb Þfqge ¼ fF g;

ð8Þ

4 disks 60.3 kg each

0.09m

Bearing 0.5

Bearing

0.5

0.5

0.5

0.5

0.5

0.5

m

Element No. (1) 1 Node No.

(2) 2

(3) 3

(4) 4

(5) 5

(6) 6

(7) 7

Fig. 2. Rotor-bearing system with disks.

8

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where ½M s ; ½G s ; ½K s are the assembled mass, gyroscopic and stiffness matrix of the system incorporating the disk mass matrix; ½cb and ½kb are the bearing damping and stiffness matrices, respectively (see Fig. 2). To determine the eigenvalues and eigenvectors the right-hand side of the equation is set to zero. The element matrices are identical to Rao [10].

3. Results and discussion 3.1. Validation A steel shaft having diameter 10.16 cm and length 127 cm supported by two identical isotropic bearings at both the ends is discretised into seven equal finite elements. The present set of results are compared with those obtained by Ku [9] and Ozguven and Ozkan [7] as shown in Table 1 and are found to be in good agreement. The whirl speed map of simple rotor systems supported on undamped isotropic bearings having stiffness coefficients Kvv ¼ Kww ¼ 1:7513  107 N/m and Kvw ¼ Kwv ¼ 0:0 are presented in Figs. 3 and 4. In Fig. 3 hysteretic damping is considered, whereas viscous damping is considered in Fig. 4. It has been observed that for the shaft material with viscous internal damping Zv ¼ 0:0002 s, critical speeds for the first and second forward modes are found to be 5000 and 10,782 rpm, respectively in the present case, whereas these are reported as 4960 and 10,500 rpm, respectively by Ku [9]. All other results are found to be in good agreement. The damping coefficients Cvv ¼ Cww ¼ 1:7513  103 ; Cvw ¼ Cwv ¼ 0:0 are also included along with the stiffness coefficients for the damped isotropic bearings. The whirl speed maps are presented in Figs. 5 and 6. These plots are compared with those obtained by Ku [9] and found to be in good agreement. In view of the above, the model used for the hypothetical bearings is used for analysis of the rotor-bearings systems supported on fluid film bearings. 3.2. Numerical example A typical simply supported rotor disk system mounted on two identical fluid film bearings is analysed using the present finite element model (Fig. 2). The physical properties of the shaft and bearing geometry are given in Table 2. Table 1 Whirl speeds in rad/s of a uniform shaft with isotropic undamped flexible bearings at a spin speed of 4000 rpm ZH ¼ 0:0002

Zv ¼ 0:0002 s

Mode

Present work

Ref. [9]

Ref. [7]

Present work

Ref. [9]

Ref. [7]

1F 1B 2F 2B 3F 3B 4F 4B

519.4 520.0 1091.6 1094.8 2228.0 2241.9 4954.1 4987.7

519.78 519.23 1094.40 1090.90 2238.53 223.80 4968.16 4935.91

520.06 521.79 1096.01 1095.34 2222.78 2206.94 4447.40 4411.81

519.4 520.0 1091.6 1094.8 2227.9 2241.8 4933.6 4987.2

519.75 521.48 1095.13 1094.52 2216.81 2201.25 4413.32 4378.95

520.01 519.54 1095.28 1091.77 2244.72 2229.82 5020.12 4986.74

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Fig. 3. Campbell diagram for rotor-bearing system with undamped isotropic bearing.

Fig. 4. See caption of Fig. 3.

The dynamic coefficients for plain journal bearing with two axial grooves 180 apart and of 20 circumferential extent each are evaluated using the code based on the numerical scheme of Lund [11]. Initially Sommerfeld numbers are calculated for different operating conditions including speeds; then corresponding eccentricity ratios are found from the code. The code facilitates estimation of the stiffness and damping coefficients for particular eccentricity ratio. These dynamic coefficients of the bearings are used in the FEM model to estimate the natural whirl frequencies of the rotor-bearing system. Results are presented in the form of Campbell diagrams for finite bearing in Figs. 7–9. Fig. 7 presents critical speeds for first four modes for rotor with hysteretic damping. Critical speeds for rotors with viscous damping are presented in Fig. 8. Fig. 9 demonstrates the effect of both hysteretic and viscous damping. Not much difference in critical speeds is observed in all the three cases considered namely hysteretic damping, viscous damping and both hysteretic and viscous damping.

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Fig. 5. Campbell diagram for rotor-bearing system with damped isotropic bearing.

Fig. 6. See caption of Fig. 5. Table 2 Physical property of the shaft and geometry of the bearings Density of the shaft material, r Elastic modulus of the shaft material, E Diameter of the bearing, D Bearing load, W Radial clearance, C

7830 kg/m3 2.08E+11 N/m2 0.09 m 1960.0 N 0.00254 cm

3.2.1. Half-frequency whirling An interesting observation of the analysis is that in addition to the synchronous natural whirl frequencies for the first four modes, for every spin speed another whirling frequency appears in the solution as shown in Figs. 7–9. This particular frequency is around half the spin speed; and

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Fig. 7. Campbell diagram for rotor-bearing system with hysteretic damping using finite bearing stiffness and damping coefficients.

Fig. 8. Campbell diagram for rotor-bearing system with viscous damping using finite bearing stiffness and damping coefficients.

therefore this value of frequency is identified as the half-frequency whirling due to the use of fluid film bearings. It may be mentioned that half-frequency whirling or oil whirl is very common in case of fluid film bearings.

3.2.2. Short bearing approximation It has been found that many researchers use short bearing approximation to estimate the stiffness and damping coefficients of fluid film bearings [10]. The results of the system with hysteretic damping supported on fluid film bearings using short bearing stiffness and damping coefficients are also presented. Critical whirl frequencies for both finite and short bearings are estimated as shown in Table 3. For the first mode, critical speeds for both short and finite bearing are same. But from the second mode, in case of short bearings, critical speeds are more than those

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Fig. 9. Campbell diagram for rotor-bearing system with hysteretic and viscous damping using finite bearing stiffness and damping coefficients. Table 3 Critical speeds in rad/s for rotor-bearing system with hysteretic damping Mode no.

Viscosity=0.00568 Ns/m2 Finite bearing

I II III IV

Short bearing

Backward

Forward

Backward

Forward

79.03 304.83 619.29 933.87

79.03 310.52 654.38 998.24

79.03 304.83 620.96 937.09

79.03 317.54 657.89 994.73

for finite bearing case. However, for the fourth forward mode, short bearing results show decreasing trend of the frequencies. In addition to the synchronous whirling frequencies, additional frequencies for the first four modes are also plotted in Fig. 10 using short bearing stiffness and damping coefficients. It may be observed that these additional frequencies are of the same order as that of the regular synchronous whirling frequencies. But in the case of finite bearings these frequencies depicted the half-frequency whirling condition. This has shown the deficiency in the use of short bearing approximation in studies relating to estimation of whirling frequencies, both synchronous and half-frequency whirl conditions. 3.2.3. Effect of bearing damping coefficients To demonstrate the effect of the bearing damping coefficients, another Campbell diagram is presented in Fig. 11. The half-frequency whirl is not found in the absence of bearing damping coefficients as seen in the figure. Comparing this figure with Fig. 9, it has been observed that the synchronous whirling frequencies in all the four modes are not much different in absence of bearing damping coefficients.

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Fig. 10. Campbell diagram for rotor-bearing system with hysteretic damping using short bearing stiffness and damping coefficients.

Fig. 11. Campbell diagram for rotor-bearing system without bearing damping coefficients (finite bearing).

3.2.4. Effect of cross-coupled stiffness To investigate the effect of cross-coupled stiffness coefficients of the bearings, a Campbell diagram is plotted as shown in Fig. 12. In this case the bearing cross-coupled stiffness coefficients are not used for estimating whirling frequencies. Comparing Fig. 12 with Fig. 9, it can be observed that the additional frequencies in absence of cross-coupled stiffness coefficients are of much lower order. Comparing Fig. 12 with Fig. 11, it is found that in absence of bearing damping coefficients, the additional frequencies are not at all there. But in absence of bearing cross-coupled stiffness additional frequencies of lower order are present. Therefore, it can be concluded that the root cause of half-frequency whirling is bearing damping coefficients and not the cross-coupled stiffness coefficients.

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Fig. 12. Campbell diagram for rotor-bearing system without bearing cross-coupled stiffness coefficients (finite bearing).

4. Conclusion The study to find out the behaviour of rotor-bearing system mounted on two identical fluid film bearings results in some interesting observations. From the presented results it can be concluded that in addition to the natural whirl frequencies for the first four modes, for every spin speed another whirling frequency appears in the solution, which is around half the spin speed. But in case of short bearings, these additional frequencies are of the same order as that of the regular whirling frequencies. The half-frequency whirling phenomenon can be attributed due to bearing damping; as these frequencies disappear when the bearing damping coefficients are not used. The presented results may be of interest to the designers and researchers. Nevertheless, the observations made here from the theoretical analysis should have experimental verification, which was not possible at the present time due to unavailability of test rig. It would be perhaps better to carry out more rigorous, particularly experimental, investigations to characterise any rotorbearing system supported on fluid film bearings.

References [1] R.L. Ruhl, Dynamics of distributed parameter rotor systems: transfer matrix and finite element techniques, Ph.D. dissertation, Cornell University, 1970. [2] R.L. Ruhl, J.F. Booker, A finite element model for distributed parameter turbo rotor systems, Journal of Engineering for Industry ASME 94 (1972) 128–132. [3] T. Thorkildsen, Solution of a distributed mass and unbalanced rotor system using a consistent mass matrix approach, MSE Engineering Report, Arizona State University, 1972. [4] H.D. Nelson, J.M. McVaugh, The dynamics of rotor-bearing systems using finite elements, Journal of Engineering for Industry 98 (1976) 593–600. [5] E.S. Zorzi, H.D. Nelson, Finite element simulation of rotor-bearing systems with internal damping, Journal of Engineering Power ASME 99 (1977) 71–76. [6] H.D. Nelson, A finite rotating shaft element using Timoshenko beam theory, Journal of Mechanical Design ASME 102 (1980) 793–803.

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[7] H. Nevzat Ozguven, Z. Levent Ozkan, Whirl speeds and unbalance response of multibearing rotors using finite elements, Journal of Vibration, Acoustics, Stress, Reliability in Design 106 (1984) 72–79. [8] L.W. Chen, D.M. Ku, Finite element analysis of natural whirl speeds of rotating shafts, Computers and Structures 40 (3) (1991) 741–747. [9] D.M. Ku, Finite element analysis of natural whirl speeds for rotor-bearing systems with internal damping, Journal of Mechanical Systems and Signal Processing 12 (5) (1998) 599–610. [10] J.S. Rao, Rotor Dynamics, New Age International Publishers, New Delhi, pp. 146–266. [11] J.W. Lund, Rotor-Bearing Dynamics, Technical University of Denmark Lecture notes, ISBN 83-04-00267-1, 1979.