VIBRATION ANALYSIS OF ROTOR - COUPLING - BEARING SYSTEM WITH MISALIGNED SHAFTS

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1 - THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47th St, New York, N.Y GT-12.. The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of be Society or.of iisdivisions or e. Sections, or printed in its publications. Discussion is printed only d the paper is published in an ASME Journal. Authorization to photocopy. material for internal or personal use under circumstance not falling within the fair use provisions of the Copyright Act is granted by ASME to libraries and other users registered with tie Copyright Clearance Center (CCC) Transactional Reporting Service provided that the base fee of $0.30 per page is paid directly to the CCC, 27 Congress Sheet, Salem MA Requests for special permission or bulk reproduction should be addressed to die ASWE Technical Rbishing Department ' - ' - Copyright by ASME All Rights Reserved VIBRATION ANALYSIS OF ROTOR - COUPLING - BEARING SYSTEM WITH MISALIGNED SHAFTS A. SREENIVASA RAO A. S. SEKHAR ilijitio j Department of Mechanical Engineering Indian Institute of Technology Kharagpur , INDIA ' ABSTRACT The shaft misalignment, even being a common fault in rotating machinery, is not sufficiently studied. The present work addresses effects of. misalignment in rotating machinery. An attempt to give a theoretical model for a rotor- coupling -bearing system has been done. The. 'rotor-bearing system including the flexible coupling is modelled using the finite elements. The reaction forces and moments developed due to flexible coupling misalignment both for parallel and angular are derived and introduced in the model.' Vibration analyses such as eigen value analysis and unbalance response are carried out for the rotor system with misaligned shafts. NOMENCLATURE (Cl matrix includes gyroscopic effects and damping elastic modulus FX1,FX2... reaction forces diametral inertia Ka axial spring rate per disc pack y "b bending spring rate per degree per disc pack K c flexible coupling stiffness matrix (K) stiffness matrix E element length [MI mass matrix {0. vector for nodal quantities (0) force vector time Tq torque AX, AY... misalignment displacements AZ stretch(+) or compression(-) of complete coupling from its free length 23 centre of articulation 0 spin angle misalignment angle 0 1 sin (AX 1 / 23) 43 1 sin 2 sin (AY 1 / 23) (AX 2 / 23) IP 2 sin -1 (AY 2 / 23) (7)d' t d ) mass centre location in disc spin speed derivative with respect to time 1. INTRODUCTION In industrial rotor-bearing systems the rotor unbalance and shaft misalignment are the two main sources of vibrations and are major concerns. A number of analytical methods have been applied to unbalance response calculation and is well understood, unlike shaft misalignment. A recent review on couplings by Xu and Downloaded From: Presented at the International 10/26/2018 Gas Terms Turbine of Use: and Aeroengine Congress & Exhibition Birmingham, UK June 10-13, 1996

2 Marangoni (1990) suggests that further. studies on the vibration induced by misaligned flexible coupling, as well as frequency detection techniques are needed to understand clearly the mechanisms of the system misalignment. Bloch (1976) identified the forces and moments developed by a misaligned gear coupling. Gibbons (1976) showed that these forces and moments are developed by different types of misaligned couplings. Comparative values of these forces were presented. The effect that these forces have upon the machines has been described in general terms. However, little work has been done to incorporate an accurate flexible coupling model into the system analysis. Very recently some attempts have been done in this regard (Simon, 1992 ; Xu and Marangoni, 1994a,b; Chaika, 1994 ; Sekhar and Prabhu, 1995). Xu and Marangoni (1994a,b) in recent papers presented theoretical model and analysis using mode synthesis together with experimental results considering the universal joint effect for the motor - flexible coupling - rotor system capable of describing the mechanical vibration resulting from misalignment and unbalance. In an other recent paper Sekhar and Prabhu (1995) developed rotor - coupling - bearing system using higher order FEW and introduced the reactive forces and moments at the coupling position to simulate misalignment conditions. However, the flexibility of coupling is not considered in that paper. Also the more accurate but more time consuming (CPU) higher order FEM is not so much required for the rotor system with misaligned shafts. The present analysis modifies this work by introducing the model of a flexible coupling using FEM with conventional beam elements. 2. SYSTEM EQUATION OF MOTION Considering two translational and two rotational degrees of freedom for bending mode, at each node (q 1 - cis per element) the equation of motion of the rotor system is of the form IMMO + [C]ic + W{g) = {(2) (1) The details of the individual matrices of eq.(1) are given in (Nelson and McVaugh, 1976). The details of flexible coupling matrices are discussed in section 3. The eq.(1) is extended to include the axial and torsional modes. The axial and torsional mode shape functions are derived by assuming a linear variation along the z axis of the element to obtain six degrees of freedom (Rao, 1980). No coupling between the axial, bending and torsional modes is assumed while deriving the matrices. However, the analysis here is done for bending response only. For computational purposes, eq.(1) is written in the first order state vector form and can be solved for the eigenvalues. The unbalance response of the system is solved by assuming a harmonic solution as done in Sekhar and Prabhu (1995). The unbalance response force for eq.(1) upon assuming in two harmonics can be of the form {(2) = {Q C }[cos Qt + cos 20t] + where (0 ) 7 = ( n d {Q [sin Qt. + sin 2Qt] (2) 0 ol; (Q } 7 d = 1-C n d d o and hi, ) is the mass centre location in d d the disc. A steady state solution of the similar form, icil = {Mu's sit + (ci s Isin Qt. + (qdcos 2Qt. + (cidsin 2Qt (3) is assumed and substituted in eq(1), which yields the solutions in two harmonics as, icici) UM -0 2 [1.1]) n [C] 1 - n [C] ([K]-0 2 [m]] {. -1 (4) 1{gcl [UK] [m]) 2Q [C]i {{(211. (qs2 ) -20 [C] ([K] [M]) s...(5) 2

3 8 "" FIG.1 CO-ORDINATES OF TWO JOINED ELEMENTS ( from Kramer, 1993) 3. MODELLING OF FLEXIBLE COUPLING The modelling of flexible coupling as 'discussed in Kramer (1993) is used in the present study. The separate shafts of a complete line-out can be connected together by either rigid or flexible couplings. For rigid couplings the connection can be treated as a part of the shaft or as a disc and will be assembled in the system equation (1). However when a flexible coupling is used two types of technical possibilities arise. (i) a frictionless joint or (ii) a joint with stiffness and damping. For a frictionless, radially stiff joint the displacement at the two sides of the joint are identical and angles are different. Fig. la shows the coordinates of two shaft elements which are to be joined. with q9= q s and q10= cl6 the new coordinates are shown in the Fig. lb for the elements when joined together. For example the following gives the stiffness matrix of the flexible coupling. El K = c Symmetric A M M W A ?%6I (b),,.. 11 In the second possibility, namely the joint with stiffness and damping, the moments and their rotations and time derivatives of rotations need to be incorporated as discussed in Kramer (1993). However, the present study utilizes the first possibility and incorporates at the respective nodes the forces and moments due to the coupling misalignment which is discussed in the following section. 4. COUPLING MISALIGNMENT Shaft misalignment is a condition in which the components that are coaxial by design are not actually coaxial, due either to assembly errors or to deformation of sub-units and/or their functions. There are two baiic types of shaft misalignment, parallel and angular as shown in ( Xu and Marangoni,1994a). Misalignment of machinery shafts causes reaction forces to be generated in the coupling which in turn affect the machines and are often a major cause of vibration. Hence flexible couplings are necessary to connect the turbomachines to their drives or loads. However, it is important to realize that flexible coupling can only increase the ability of a drive train to tolerate misalignment but, for high speed rotating machines, flexible coupling are not a cure for severe alignment problems. The reaction forces and moments developed due to parallel misalignment are given in Gibbons (1976). Sekhar and Prabhu (1995) developed similar forces and moments in the case of angular misalignment also, and introduced in the rotor-bearing system, to simulate the coupling misalignment. The details are repeated here as these are introduced even in the present work. Fig. 2 depicts two machine shaft centre lines, 21 and 22, which are misaligned. The centreline of the coupling spacer is shown connecting the two shaft centrelines with intersection points being the coupling centres of articulation, not the shaft's ends. For an existing machine, the values and directions of displacements AX1, AY1, AX2 and AY2 of Fig. 2 can be readily 3

4 Mut Y2 AXIS FY2 MY2A FX2 Z2AXIS (a) Parallel Misalignment 1.1 Axis FYI NX2 X2 AXIS Y1 AXIS Angular Misalignment: MX1= 0.0 ; MX2= - Kb 8 MY1= 0.0 ; MY2= Tq Sin = Tq/ Cos 0 ; MZ2= -Tq FX1 = (-MY1 - MY2) /23 ; FX2 = -FX1 FY1 = (MX1 + MX2 ) /23 )/Cos 0 ; FY2 = -FYI FZ1 = (Ka AZ + Ka (A2) 3 ; FZ2 = FZ1 In the present analysis, the linear spring rates for the flexure coupling in both bending and axial nodes are assumed. Hence the term (AZ) is neglected in the above expressions, for the following analysis.. MU 22 AXIS Y2AX IS F aj) MY 2 F X2 (IA Angular Misalignment c f Nc... FY1 MX2 X 2 AXIS 'N., An XI AXIS FIG. 2. COUPLING CO-ORDINATE SYSTEM obtained from a graphical plot of reverse indicator readings. For a machine in the planning stage, realistic values can be chosen for comparative calculations (Gibbons, 1976). Assume that 21 is the axis of the driving machine, that (+) torque is applied as shown in Fig. 2 and that rotation is in the same direction as applied torque. The reaction forces and moments which the coupling exerts on the machine's shafts are as follows. Parallel Misalignment: MXI= Tq Sin 01 + Kb 4,1; MX2= Tq Sin82-Kb (I)2 MY1= Tq Sin 4)1 - Kb 01; MY2= Tq Sint2+Kb 82 M21= Tq ; MZ2= -Tq FX1 = (-MY1 - MY2) /23 FY1 = (MX1 + MX2 ) /23 F21 = Ka AZ + Ka (A2) 3 ; FX2 = -FX1 ; FY2 = -FY1 ; FZ2 = FZ1 F 21 S. RESULTS AND DISCUSSIONS In the present analysis the study has been carried out for eigenvalues and unbalance response analyses. Rao (1983) considered a rotor system with a spline coupling as shown in Fig.3 and did analysis using transfer matrix method. In such case, the boundary conditions at the coupling considered are : the deflection is continuous, bending moment is zero, the shear force changes because of the mass at the coupling station and the slope has a jump. For the same rotor finite element analysis has been carried out using the flexible coupling model (as explained in section 2 and 3). The comparative table 1. shows a good agreement for the considered case of critical speeds without gyroscopic effects. r rr i I e kg m kg 6.69 cm 7.62 cm Mg 0.70 kg Cm LOW 7 I 27 C el. Mal shalt kg CM 7.62 Cm Spline Support Puiity toketilop FIG. 3. A TWO ROTOR SYSTEM COUPLED BY A SPLINE (from Rao, 1983) 2 4

5 TABLE 1. Bending Critical speeds of two rotor system with flexible coupling (Fig.3) Critical Speeds (r.p.m) Mode Transfer Matrix FEM Using the flexible coupling modelling and misalignment forces and moments, analysis has been carried out on hypothetical rotor as was considered in Sekhar and Prabhu (1995), where the coupling was considered as rigid only and misalignment forces were introduced at the corresponding nodes. For comparison purpose; the same rotor (Fig.4) has been considered in the present study. The mode shapes corresponding to both the cases of coupling are also shown in the Fig. 4. The following data (same as in Sekhar and Prabhu(1995)) has been used for the analyses of critical speeds (Table 2), mode shapes (Fig.4) and unbalance response for the case of with and without coupling misalignment. The bearingg stiffness and damping values are 10 N/m, 0.5 x 10 Ns/m respectively at the ends and 10 7 N/m, 1.0 x 10 3 Ns/m in the middle. The discs are having unbalance eccentricity of 0.01 mm. The data (Gibbons,1976) considered for the machine including the coupling are as follows : Power = 6500 HP/7500 rpm ; A constant Torque of Nm Diaphragm coupling : Constant thickness, Multi-Disc, Convoluted, 0.D = mm,1/2 Degree rated Misalignment Angle Centre of articulation = mm k b = 237 Nm/Deg per disc pack K a = 1260N/mm per disc pack Axial deflection (compressed) = 1.27 mm 14 finite beam elements, 15 nodes. Misalignment : parallel Angular AX1= mm AY1= mm; 0 = 0.2 AX2= mm AY2 = mm TABLE 2. Bending Critical speeds of two rotor system with rigid and flexible coupling (Fig.4) Critical Speeds (r.p.m) Mode rigid flexible Cm _3 L 07.5cm FIG. 4. ROTOR-COUPLING-BEARING SYSTEM WITH MODE SHAPES rigid ---- flexible coupling The analysis is done for both parallel (Fig.5) and angular (Fig.6) misalignment cases for the rotor (Fig.4) at the left disc. The periodic misalignment forces and moments in each case are evaluated as discussed in section 4 and are included at the appropriate degrees of freedom in rotor assembly. Though the eq.(1) is modified to include axial and torsional modes, the analysis is done for bending response only and decoupled from other modes. The bending response is of circular orbit, since the bearings considered are isotropic in stiffness. Referring to the Table 2, the peaks for the second harmonic response in both the figs. 5 & 6 are occurred at one half the first critical speeds due to the assumptions of eqs.(2&3). However, it can be noticed from the Figs. 5 & 6, that the response significantly changes with misalignment in the case of 2x vibration, which are well known from experimental and many earlier findings (Dewell and Mitchell, 1984; Piotrowski, 1986). Similar results are shown in Sekhar and Prabhu (1995) but A- 5

6 the flexible coupling modelling is not considered in that case. The dominance of 2x component with misalignment is evident from Fig.7. The amplitude ratio being with respect to the without misalignment case at the same speed. The angular misalignment values are converted to 'mm' by using 23. The reasons for the increase in 2x component only and very little for fundamental components, can be understood easily by referring to the Fig.4 and are well explained in Sekhar and Prabhu (1988) In the present analysis, the coupling considered is a flexible element coupling. In this type of coupling, the net reaction forces at the articulating points are due to the moments that in turn are due to both the angular and parallel misalignment (2TqSin0/23). In this type of coupling, the net moment due to the flexing of the element of the coupling is nullified, since the components act in opposite directions. The net reaction forces acting at the articulating points, however, are varying periodically with a period (n/w). Hence the fundamental harmonic forcing function (of period, 2n/w) due to misalignment is not present (not significant). But in the case of other types of couplings, where the spacer is not present, the reactive moments due to the angular misalignment will give rise to a reaction moment at the fundamental frequency. Hence the fundamental components appear in the responses. The comparative study of response of rotor system with flexible coupling with that of rigid coupling is shown in Fig. 8. (c0 'Co (a) 4 ISO C 00 1 so I I / I / SO a --- IOW U M 'COO Spelled Iron, Speed Ir. p.m.) '3100 a SOO 1000 IWO Speed Er. pm.) FIG.6 UNBALANCE RESPONSE WITH PARALLEL MISALIGNMENT Cc) 1 AmplitUde; (P) 2 Amplitude wichour misatignmtnt misalsoromsest 0 SOO Speed Ir.p tn.) FIG. 6 UNBALANCE RESPONSE WITH ANGULAR MISALIGNMENT (a) 1 Amplitude ; (b) 2 'Amplitude --without misalignment, with misalignment 6

7 Ol 1110 Op (a) OS , SO t. o. n Speed I o D. rol n ( b) s Of I MillalIgnmont (Wm) Misalignment IMml FIG.7 VARIATIONS OF VIBRATION AMPLITUDE RATIO WITH MISALIGNMENT (a). ( LI) Parallel misalignment and (0. (0) Angular misalignment ; IX, 2% 1 Mapliiule I p ail And as expected due to flexibility, the shift in peaks to lower speeds and rise in response with flexible coupling, are observed. The results are expected and are well understood. However, the aim of the present study is to model the rotor system and simulate misalignment. For better results, as did in Chaika (1994) for centrifugal and standard couplings, the dynamic modelling of flexible couplings needs to be incorporated in the rotor-bearing system. Similarly more simulation models and theoretical analysis on coupling misalignments, such as the universal joint effects etc. (Xu and Marangoni, 1994a,b) are essential in this field where limited work has been done. 6. CONCLUSIONS Finite element model for a rotor-flexible coupling-bearing system has been presented. The reaction forces and moments developed due to flexible coupling misalignment both for parallel and angular are derived and introduced in the model to simulate misalignment. Vibration analyses FIG. e spool (np.roi UNBALANCE RESPONSE WITH RIGID AND FLEXIBLE COUPLINGS. (a) Without misalignment ( lif/ With misalignment ; rigid coupling flexible coupling such as eigen value analysis and unbalance response are carried out for the rotor system with misaligned shafts. The lx bending vibration response shows that the coupling misalignment does not significantly alter the amplitude. But the 2x vibration response clearly shoots up with misalignment. For a machine in the planning stage, realistic values of indicator readings can be used in the present model to estimate the effects of misalignment on vibration response. For better results, the dynamic modelling of flexible couplings needs to be incorporated in the rotor-bearing system. Similarly more simulation models and theoretical analysis on coupling misalignments are essential in this field where limited work has been done. 7

8 REFERENCES Bloch, H.P., 1976, "How to uprate turbomachinery by optimized coupling selection", Hydrocarbon Processing, Vol. 55, No.1, pp Chaika, V., 1994, "Steady state response of a system of rotors with various types of flexible couplings", Journal of Sound and Vibration, Vol. 177, No.1, pp Dewell, D.L, and Mitchell, L.D., 1984, "Detection of a Misaligned Disk Coupling Using Spectrum Analysis," ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 106, pp Gibbons, C.B., 1976, "Coupling misalignment forces", Proceedings of the fifth Turbomachinery Symposium, Gas Turbine Laboratories, Texas A & M University, College Station, Texas, pp Kramer, E., 1993, Dynamics of Rotors and foundations, Springer-Verlag, Berlin, pp elastic foundations due to unbalance and coupling misalignment," Proceedings of the Institution of-mechanical Engineers, Part C : Journal of Mechanical Engineering Science, Vol. 206, pp , Xu, M., and Marangoni, R.D., 1990, "Flexible couplings : study and application," Shock and Vibration Digest, Vol. 22, No. 9, pp Xu, M., and Marangoni, R.D., 1994a, "Vibration analysis of a motor-flexible coupling - rotor system subject to misalignment and unbalance, part I : theoretical model and analysis," Journal of Sound and Vibration, Vol. 176, pp Xu, M., and Marangoni, R.D., 1994b, "Vibration analysis of a motor-flexible coupling - rotor system subject to misalignment and unbalance, part II : experimental validation," Journal of Sound and Vibration, Vol. 176, pp Nelson, H.D., and McVaugh, J.M., 1976, "Dynamics of Rotor- bearing systems using Finite elements", ASME Journal of Engineering for Industry, Vol. 98, pp Piotrowski, J., 1986, Shaft Alignment Handbook, Marcel Dekker, Inc, New York and Basel. Rao, J.S., 1983, Rotor Dynamics, Wiley Eastern limited, New Delhi, pp Rao, S.S., 1980, The finite element method in Engineering, Pergamon press. Sekhar, A.S,.and Prabhu, B.S., 1995, "Effects of coupling misalignment on vibrations of rotating machinery," Journal of Sound and Vibration, Vol.185, pp Simon,G., 1992, "Prediction of vibration behaviour of large turbomachinery on 8