Thermal Bending of the Rotor Due to Rotor-to-Stator Rub

Thermal Bending of the Rotor Due to Rotor-to-Stator Rub Bently Rotor Dynamics Research Corporation, 1711 Orbit Way, Bldg. 1, Minden, NV 89423, USA (Received 5 September 1997;In final form 21 July 1998) The rotor thermal bending due to the rotor-to-stator rubbing can lead to one of three types of observed rotor lateral motion: (1) spiral with increasing amplitude, (2) oscillating between rub]no-rub conditions, and (3) asymptotical approach to the rotor limit cycle. Based on the machinery observations, it is assumed in the analytical part of the paper that the speed scale of transient thermal effects is considerably lower than that of rotor vibrations, and that the thermal effect reflects only on the rotor steady-state vibrational response. This response would change due to thermally induced bow of the rotor, which can be considered to slowly vary in timefor the purpose of rotor vibration calculations. Thus uncoupled from the thermal problem, the rotor vibration is analyzed. The major consideration is given to the rotor which experiences intermittent contact with the stator, due to predetermined thermal bow, unbalance force, and radial constant load force. In the case of inelastic impact, it causes an on]off, step-change in the stiffness of the system. Using a specially developed variable transformation for the system with discontinuities, and averaging technique the resonance regimes of motion are obtained. These regimes are used to calculate the heat generated during contact stage, as a function of thermal bow modal parameters, which is used as a boundary condition for the rotor heat transfer problem. The latter is treated as quasi-static, which reduces the problem to an ordinary differential equation for the thermal bow vector. It is investigated from the stability standpoint. Keywords: Rotor, Vibrations, Thermal bow, Rub, Averaging technique 1. INTRODUCTION Rotor-to-stator rub, an unwelcome contact between rotating and nonrotating elements of a machine, can be one of the most damaging malfunctions of rotating machinery. Generated by some perturbation of normal operating conditions that causes an increase of rotor vibration level, and/ or an increase of the rotor centerline eccentricity, the rub can maintain itself, and gradually become more severe. The self-generating feature of this phenomenon originates from the interaction between rub-related thermal effects and lateral vibrational response of the rotor. Starting from Corresponding author. Tel." (702) 782 3611. Fax: (702) 782 9236. E-mail: psg@bncl 17psg.bently.com. pioneering works of Taylor (1924) and Newkirk (1926), the unwinding spiral vibrations of rotors are documented in several papers (Black, 1968; Kroon and Williams, 1939; Dimaragonas, 1973; Kellenberger, 1979; Natho and Crenwelge, 1983; Hashemi, 1984; Smalley, 1987; Muszynska, 1993). In addition to the spiral response, Dimaragonas, (1974) described an oscillating mode of shaft vibration, occurring during the transition from the spiraling to a steady-state mode. A similar result from an improved rotor dynamic model was obtained by Muszynska (1993). The problem of rub-related heat distribution was discussed by several authors, for example, Sweets (1966), Kellenberger (1979) and Smalley (1987). The most complete analysis of the heat transfer problem associated with rub is given in the book by Dimaragonas and Paipetis (1983). In all referenced literature the analysis of the shaft bow, resulting from the uneven temperature distribution due to rub, is based on an approximation on the mean flexural rotation of one end of the shaft in relation to the other (Goodier, 1958). The idea of the discontinuous variable transformation applied in this paper for the rub dynamics analytical description appeared first in the paper by Zhuravlev (1978), and was expanded later by Petchenev and Fiddling (1992) and Goldman and Muszynska (1994a,b; 1995a,b). 2. MATHEMATICAL MODEL OF THE RUBBING ROTOR An isotropic rotor in its lateral mode motion is considered (Fig. 1). The rub (or more generally any local nonlinearity which creates a thermal effect) occurs at the shaft axial location 12. If the rotary inertia and shear stresses are neglected, the equations of motion of the rotor can be expressed as follows: m+ 0 mgfZeJ + O, where x(1, t), y(1, t) are horizontal and vertical displacements of the rotor at the axial location IEjO2(7-r),l 0l 2 v/7-1 (1) M2(/’/) Bending Monet to Moment of Inertia Ratio: I(1,) | hermai Bow Mmle L T’ Temlratut difference (1, t) x + jy (x, + jy, " FIGURE Physical model of the rubbing rotor. in the stationary coordinates, ’-x/jy; EJ is bending stiffness, m is mass linear density,/- is a vector ofthermal bending in stationary coordinates, /(l, t) v(1, t)eJ, /, is a vector of thermal bending in rotating coordinates, is rotative speed, -t + Y is a distributed unbalance vector in the coordinate system rotating with the rotor at the particular axial location l, Q- Qx +jQy is a complex vector describing linear density of the external and nonconservative forces. Assuming that the contact/no-contact situation at the axial location 12 generates a radial reaction force distribution over the rotor surface as follows: Of,, off a L/2 -L/2 2r /4/ (...) is defined for angles as a periodic with period of 2. Since the rotor velocity at the contact location can be approximated as R2, the heat rate density g(l, t) per unit area equals to the friction force power per unit area. Taking into account Eq. (4), it can be presented as follows: U- -OKu([T2] c) F2 172] > c o-,IFzl 0 ifl&l < c {1 if x+ (2) (9+-arg(7)). (5) The thermal conductivity equation with initial and boundary conditions where KT is the local stiffness of the stationary obstacle, Y2 7(/2, t), c is the radial clearance between the rotor and the obstacle, the vector Q can be presented in the following form" OZT / 10T 10T O2T Or t=0 + 7 012 at fiT I=0,L =0, 10T -c(1) + q(l)h() + (1 + () +r _x(,e,t), () describes heating of the shaft due to rub. Here T= T(r,,l,t) is a difference of temperatures Here.C(/) is a damping linear density, q(l), ?(1) are radial side-load forces linear density and their angular orientation, respectively, f is a dry friction coefficient, L is the length of the rotor and (... is the Dirac function. The thermal bow appears due to an uneven temperature distribution along the rotor caused by the friction force-generated heat. The latter can be characterized by the heat rate density g(l, 9, t) per unit area ofthe rotor cross section (is an angular coordinate, Fig. 2). Considering the particular area element R2dl d (R2 is the rotor external radius), on the rotor surface around the point with axial coordinate l, the friction force jfN is applied to the rotor ir CN-- deN CN+ deN and -dl +dl (it is assumed that the rotor-to& & stator contact occurs at a single location only), where 9N- 2-- + arg(&) is the angular position of the friction force (Fig. 2). The friction force, therefore, can be expressed in a form of a FIGURE 2 Cross-section of the rotor at the rub axial location between the environment and the material point of the shaft with coordinates 1, r, at an instant t, and ,/ are thermodynamic constants, R1 is the rotor internal radius. To complete the problem formulation, a relationship between the temperature distribution and thermal bow has to be derived. In order to accomplish this, the assumption is made that the rotor can be considered within the limits of Euler’s beam theory. In this case, the thermal stress-related bending moments in the rotating coordinate axes Xr, Yr are as follows: following expression for M(I, 12): n m-Z2l- (Lrn_l,,7 My-- -)T db fR ETrZ R2 cosbdr My(t, 1), Mx ETr 2 sin dr Mx( t, ), Obviously the expression -(M(1,12)/I(12)) determines the mo+de shape of the thermal bow (see Fig. 1) while i?(t) plays the role of the modal time vector-function. The equations which describe the latter can be derived from Eqs. (6) by integral transformations (see Taylor (1924) for details), with the assumption that all thermal processes are limited to the rotor area with the same cross-section defined by internal radius R1(12) and external radius R2(12), and averaged over the yet unknown period of mechanical oscillations (e is a small parameter)" where XT is the thermal expansion coefficient, E is Young’s modulus of elasticity. According to the Castigliano’s theorem and Eqs. (7), the vector fir(l, t) of the thermal bow in the rotating coordinates can be expressed in the form of convolution: dr where \. DT (1 2)[k (1 q- 32) q- /(1 2)] (1-N2)(kl-+-th) q- h lq-t2q-8N3-+-N(1--N2)- where M(1, la) is a bending moment at the axial location la, resulting from the unit load applied at the axial location 1. For simplicity it is assumed that the rotor has constant modulus of elasticity E for all cross-sections along its length. Due to the local character of the rotor heating (Fig. 1) the convolution can be approximated as shown in the second of Eqs. (8). An example of a simply supported rotor with overhung impeller, shown in Fig. 1, has the fR2(12)2 Since, according to Eqs. (8),/3 r is proportional to the temperature ;?- i?i2 di, it follows from the first Eq. (10) that d/3r/dr O(e/3r) has a higher order of smallness than fir itself. As a consequence, the thermal bow in Eq. (1) can be considered as a parameter, and the system dynamics become essentially decoupled from the thermal part of problem. The rotor lateral mode modal coordinates Uq + jVq (q 1,2,..., oo) togetherwithorthogonal modal functions % (q-1,2,..., oo) can be introduced in a way that the following equations are satisfied: as follows: ffq 2,1- c cr- ZAq q=l (1- o-A), Z 7]q(l),q(t) + e Z Z]q(l)(q -4j(p for statically "loose" case (no contact), -1 for statically "tight" case (contact), k=l k=l (l,/2) dl q(l) M1 1(/2) rCq. (12) In this case, Eq. (1) can be written in modal coordinates as follows: where cA is an absolute value of a gap between the rotor static position and the stator. According to physical assumptions, A << 1. The coefficient A is used below as a measure of smallness. (2) One mode in particular, the kth mode, for example, governs the contact at the axial location 12. It means that this mode delivers a much higher contribution into [F(12, t)] than all other modes: o(1), ]q(/2) Here Mq, Dq and Kq are modal mass, damping and stiffness of the qth mode, q mTq dl is a modal unbalance, PqeJTq qeJ/lq dl is a modal radial side-load force. The analysis of Eqs. (13) which represent the mathematical model of the system, is performed below under the following physical assumptions: //q O(x/-) (q In almost all practical situations this is true. (3) The vertical (imaginary) direction for each mode is chosen opposite to the direction of the corresponding radial side-load force: ")/q-3rc/2, q- 1,2, k). New variables are introduced as follows" (1) The system of applied radial side-load forces maintains the static position of the rotor very close to the stationary obstacle at the axial location 12. The rotor-to-stator contact is intermittent due to the dynamic action of the unbalance. (q#k), P/e jzk --jc Kk (1 Ek A/), (14) (q -j /Xq Nq (16) where (u+jv) and rq (qk) are dynamic components of Eqs. (13) solutions, h is the In this case, the absolute value of the shaft static displacement at the rub axial location is P. GOLDMAN et al. nondimensional distance between the rotor and an obstacle at the axial location 12, of smallness, but the conditions (19) of the switch are precise. s- l(12, t)]/(c/ ) 0(1), or- arg((12, t)). 3. MECHANICAL RESONANCES AND With the new set of variables h, u, Iq (q- k) using Eqs. (16) and (13) can be rewritten as follows: HEAT GENERATION The rotor resonance responses to the unbalance which are considered below are associated with the leading mode, or in mathematical terms, are described by the first two equations (17). It means that the rest of the Eqs. (17) determine only "forced" solutions, which are defined at each sequential approximation by the previous approximation to the solution of the first two equations. Therefore, it is important to consider the latter in h" + n(1 -t- @2)h crn AH + O(A2), F utt + nZqFq + o( x :) 2 2 -jOcrnpqh + x/q[aqe j(+cq) -+-7rq, sTeJ(+cv)] + zx + where (17) -2nkh’ sin(g) + c) + sr(er/, + 1) sin(g) + cr) + n2k U- -2nu + eca cos(g) + a) + /r,srcos(g) + at) + Opngh Wq ’q -2gqnqE + Opqnqah u + Dk,q u2 u’2 + Z (n qv/:k n2q)Im r q], (18) U,q 2Mk,qUk,q Kk,q 2 mk,q nk,q U,q Jk akq e,q c2 O(1) O(1) t, "’" gf p,q gf k’q p2 Kk,q -k l’2k-l-/kZ 7rq, 7rq" cA27rq, STejecT, --Trq I(/2) M1 (/2,12). generating approximation, which is defined by neglecting all right-side terms. The second of Eqs. (17) in generating approximation has a simple solution: u= pcos0, p’ =0, 0’= nk, The first of Eqs. (17) in generating approximation, together with the switch conditions (19), is more complex. Its solution can be built using piecewise integration, and connecting The conditions (3) of the contact/no-contact switch now become very simple: 0- f0 ifcrh>0, ifcrh <_ 0. (19) Note, that the right-side terms of Eqs. (17) are calculated with an accuracy up to the second order conditions at the ends of continuity intervals: l+x()p2 l+x()p2 1-5 sin cos (20) where S is constant referred as an amplitude parameter, where Q, F, U, W are functions of new variables, is determined by the system of Eqs. (17) and an integration variable. Equations (22) have the required format, with three fast rotating phases 0, and two slow variables S and p. This allows application of the Averaging Method [7]. Note that at this point, the equations are limited to the terms of the first order of smallness, and only small ., -1 (21) The relation between h and S (amplitude parameter) and (phase) is shown graphically in Fig. 3. It also shows the relation between the amplitude parameter S and overall amplitude A. Equations (20) and (21) constitute a variable transformation. This transformation allows to introduce the rotor vertical response amplitude parameter S and and is used to convert the original "phase" system (17) into the form with three rotating phases 0 and two slow variables S, p: right-sided terms of Eqs. (22) have discontinuities, as they change with the fast rotating phase As it results from the expression (21), the ratio co/nk is contained within the following limits: ,, S’ <--< co -n:-1 + 1/X,/’I _+_p2 if or- +1, ifcr--1. ’ p,_ O, p, ’q(S, )) + O(A2); co coAF(S, p, O, ,’q(S, )) + O(A 2) A U(S, p, O, rq(S, ,)) sin O AQ(S, p, , , ,, , nk + l/V/1 co <--<v/l+ p2 +p2- nk (24) + O(A2); 1-n-0, 1-ico-0 (i- 1,2,3,...), 2n co 0 (for cr -1). 0t- II k zxu(s, p, o, r (s, ))cos0 O(A 2) pnl (22) ’--1 and a number of noncritical variables These inequalities, together with Fourier analysis of the right-side terms of Eqs. (22), show that possible resonances occur when: (25) rq _jcrp2q nq sin fo0 + O(zX) (23) The analysis of Eqs. (22) from the standpoint of balance between the supplied and dissipated energies, allows the stationary resonance solutions for ic 0 (i-- 1,2, 3,... ), or the case of nk 0, for the case of a combinational resonance: n 0 and 2n-co=0. The first resonance occurs when the rotative speed of the shaft f2 is close to the rotor A) Normal-loose case S-I- or-1 B) Normal-tight case contact no contact no contact contact FIGURE 3 Variable nondimensional distance h (c- 721)/c/X as a function of phase leading mode natural frequency vk. It could either be accompanied by the vertical resonance 2nk-a--0 or not. This regime, referred to as a horizontal mode resonance, is described in detail in Taylor (1924). Important thing to know about horizontal mode resonance is that, due to the symmetric heating, the thermal bow does not occur. The sequence of resonances 1-ico(Sa)=O (i= 1,2, 3,... ), referred to as vertical mode resonances, is of interest because it creates uneven heating which results in a thermal bow. The resonance frequency equation according to Eqs. (21) determines the zeroth approximation Sa to the slow variable S as a function of the ratio f/iu in the corresponding resonance zone. A simple analysis of Eqs. (21), together with inequalities (24), shows that for each value of 1,2,... it can be satisfied only within the range of rotative speeds determined by the following inequality: u of the leading kth mode, the subsynchronous 1/2 regime (i 2) occurs at rotative speeds higher then 2u, and so on. In the case of normal-loose situation, (or-+ 1) the maximum rotative frequency of the corresponding resonance regime is lower then 2iu, while in the normal-tight situation 2iuk is a minimal rotative frequency. This agrees with practical observations of rubbing rotor behavior (Choi and Noah, 1987). The parameter p, which affects the width of the frequency band for each regime, characterizes the stiffening effect of the rotor-to-stator contact. Equations (22), after averaging in proximities of vertical resonances (see details in Zhuravlev (1978)), allow for the following stable stationary solutions: + sign(G) arccos bp2 i- 1,2,3,... :r 7rk(1 @/1 n)[(1 +p )nk 1]qkG 1/Sa COSO + ncr/SasinO)’ arccos G(Sa,p) This means that the main, 1 (synchronous) regime of rotor vibrations (i- 1) occurs at rotative speeds higher then unaltered resonance frequency Sa + 7r(1 p2 in: + P)Cr/1 (27) where Sa is a function of the rotative speed, determined by the solution of resonance equation ico(Sa) 0 (i 1,2, 3,...), /- be j kak e j"k + ST(kTrk, + 1)e j"T is an equivalent vector of unbalance which includes a component due to the thermal bow. It defines the series of x, 1/2x, 1/3 x,... regimes in the rotative speed bands roughly described by the inequality (26). The relationship of the vertical response phase ( -t- O/i) and overall amplitude A with rotative speed ratio for the vertical modes, calculated for a particular set of parameters or, p, and b, is presented in Fig. 4. Based on the resonance solutions (27) and expressions (11), the heat generating vector () can be calculated: where Sa is determined by the second of Eqs. (25) and Eqs. (21) as a function of rotative speed f. Since ()) is essentially the forcing function in Eq. (10) for the thermal bow, its behavior determines the behavior of the thermal bow. Figure 5 depicts ()) in a polar plot format for different resonance regimes i= 1,2, 3, 4. ’ 4. SUMMARY AND CLOSING REMARKS This paper outlines the modeling of thermal/ mechanical effects of one of the most destructive malfunctions in rotating machinery: the rotorto-stator rub. The thermal/mechanical problem is partially uncoupled by the assumption that the thermal process is relatively slow. As a result, the rotor thermal bow remains in the mechanical equations as a parameter which can be considered a constant. The combination of the Averaging Method and the assumption that the thermal processes are quasi-static allows the heat transfer problem to reduce to a vectorial ordinary differential equation (Eq. (11)), with the heat generating equivalent vector as a forcing function. This equation allows a realistic estimate of the thermal <) jf fcAKf cr V/Sza + Pz(SZa 1) 27ri(1 e -2jo) / p2) jcos(1 +ngl? +p2) +jcos [1--e-2J9(cos26--jnkV/l+p2sin2(5)]}e j7, [ 00 Rotative speed versus k-th mode natural frequency ratio 3O Rotatlve speed versus k-th mode natural frequency ratio FIGURE 4 Phase and nondimensional amplitude of the rotor-to-stator distance versus rotative speed to natural frequency ratio for the series of Ix, 1/2x, 1/2x, 1/4x resonances. Calculations are made for the parameters cr-- 1, p 2, b/ 4. FIGURE 5 Heat generating vector (} versus rotative speed to the natural frequency ratio in the polar plot format. Heavy pot indicates the position of equivalent unbalance vector b. cr-- 1, p--2, b/ 4. Goldman, P. and Muszynska, A. (1994a), Dynamic effects in mechanical structures with gaps and impacting: order and chaos, Transactions of ASME, Journal of Vibration and Acoustics, 116, 541-547. Goldman, P. and Muszynska, A. (1994b), Resonances in the system of the interacted sources of vibration. Formulation of problem and general results, International Journal for Nonlinear Mechanics, 29(1), 49-63. Goldman, P. and Muszynska, A. (1995a), Rotor-to-stator-rubrelated, thermal/mechanical effects in rotating machinery, Chaos, Solitons, and Fractals, Vol. 5, No. 9. Goldman, P. and Muszynska, A. (1995b), Smoothing technique for rub or looseness-related rotor dynamic problems, Proceedings of 1995 Design Engineering Conferences, Vibration of Nonlinear Random and Time-Varying Systems, DEVol. 84-1. Goodier, J.N. (1958), Formulas for overall thermoelastic deformation, Proc. 3rd Int. Congr. Appl. Mech., John Wiley, New York, p. 343. Hashemi, Y. (1984), Vibration problems with thermally induced distortions in turbine-generators rotors, vibrations in rotating machinery, Third Int. Conf. Proceedings, IMechE, C271/84, York, U.K. Kellenberger, W. (1979), Spiral vibrations due to the seal rings in turbogenerators. Thermally induced interaction between rotor and stator, ASME Paper 79-DET-61, Design Eng. Tech. Conf., St. Louis. Keogh, P.S. and Morton, P.G. (1993), Journal bearing differential heating evaluation with influence on rotor dynamic behavior, Proceedings of the Royal Society, London, A441. Kroon, R.P. and Williams, W.A. (1939), Spiral vibration of rotating machinery, Proc. 5th Int. Congr. Appl. Mech., John Wiley, New York, p. 712. Muszynska, A. (1989), Rotor-to-stationary element rub-related vibration phenomena in rotating machinery, literature survey, The Shock and Vibration Digest, 21(3), 3-11. Muszynska, A., Franklin, W.D. and Hayashida, R.D. (1990), Rotor-to-stator partial rubbing and its effects on rotor dynamic response, The Sixth Workshop on Rotordynamic Instability Problem in High Performance Turbomachinery, NASA CP 3122, College Station, Texas, pp. 345-362. Muszynska, A. (1993), Thermal/mechanical effect of rotor-tostator rubs in rotating machinery, Vibration of Rotating Systems, DE-vol. 60, ASME Design Technical Conf., Albuquerque, New Mexico. Natho, N.S. and Crenwelge, O.E. (1983), Case history of a steam turbine rotordynamic problem: Theoretical versus experimental results, Vibration Institute Proceedings, Machinery Vibration Monitoring & Analysis, pp. 81-89. Newkirk, B.L. (1926), Shaft rubbing, Mech. Eng., 48, 830. Petchenev, A. and Fiddling, A. (1992), Hierarchy of the resonant motions of vibroimpacting system excited by the inertia source with limited power (in Russian), Mechanika Tverdogo Tela, No. 4. Smalley, A.J. (1987), The Dynamic response of rotors to rubs during startup, Rotating Machinery Dynamics, Vol. 2, Eds. A. Muszynska and J.C. Simonis. Sweets, W.J. (1966), Analysis of rotor rubbing, GE Technical Information Series, DF-66-LS-70. Taylor, H.D. (1924), Rubbing shafts above and below resonant speed, GE Technical Information Series. No. 16709. Zhuravlev, B.F. (1978), The equations of motion for the systems with ideal one sided restrictions (in Russian), PMM (Applied Mathematics and Mechanics), Vol. 42, No. 5, pp. 781-788. bending mode (Fig. and Eqs. (9)), and can be applied not only to rub-related heating. The other possible application is an effect of journal bearing differential heating, described in the paper by Keogh and Morton (1993). The heat generating equivalent vector for the rub is calculated by application of the discontinues variable transformation and resonance version of the Averaging Method to the mechanical part of the problem. Based on the heat generating equivalent vector behavior, predictions can be made on the one of three possible thermal bow behaviors: Asymptotic approach to the equilibrium state of the thermal bow. Increasing spiraling motion of the thermal bow in the direction opposite to rotation. Slow oscillations of the thermal bow. The analytical algorithm described in this paper (Eqs. (20), (21), (23) and (27)) has a high potential as a valuable research and prediction tool for investigating rub and thermal effects in rotating machinery. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Rotating Machinery Hindawi Publishing Corporation

Thermal Bending of the Rotor Due to Rotor-to-Stator Rub

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Abstract

Bently Rotor Dynamics Research Corporation, 1711 Orbit Way, Bldg. 1, Minden, NV 89423, USA (Received 5 September 1997;In final form 21 July 1998) The rotor thermal bending due to the rotor-to-stator rubbing can lead to one of three types of observed rotor lateral motion: (1) spiral with increasing amplitude, (2) oscillating between rub]no-rub conditions, and (3) asymptotical approach to the rotor limit cycle. Based on the machinery observations, it is assumed in the analytical part of the paper that the speed scale of transient thermal effects is considerably lower than that of rotor vibrations, and that the thermal effect reflects only on the rotor steady-state vibrational response. This response would change due to thermally induced bow of the rotor, which can be considered to slowly vary in timefor the purpose of rotor vibration calculations. Thus uncoupled from the thermal problem, the rotor vibration is analyzed. The major consideration is given to the rotor which experiences intermittent contact with the stator, due to predetermined thermal bow, unbalance force, and radial constant load force. In the case of inelastic impact, it causes an on]off, step-change in the stiffness of the system. Using a specially developed variable transformation for the system with discontinuities, and averaging technique the resonance regimes of motion are obtained. These regimes are used to calculate the heat generated during contact stage, as a function of thermal bow modal parameters, which is used as a boundary condition for the rotor heat transfer problem. The latter is treated as quasi-static, which reduces the problem to an ordinary differential equation for the thermal bow vector. It is investigated from the stability standpoint. Keywords: Rotor, Vibrations, Thermal bow, Rub, Averaging technique 1. INTRODUCTION Rotor-to-stator rub, an unwelcome contact between rotating and nonrotating elements of a machine, can be one of the most damaging malfunctions of rotating machinery. Generated by some perturbation of normal operating conditions that causes an increase of rotor vibration level, and/ or an increase of the rotor centerline eccentricity, the rub can maintain itself, and gradually become more severe. The self-generating feature of this phenomenon originates from the interaction between rub-related thermal effects and lateral vibrational response of the rotor. Starting from Corresponding author. Tel." (702) 782 3611. Fax: (702) 782 9236. E-mail: psg@bncl 17psg.bently.com. pioneering works of Taylor (1924) and Newkirk (1926), the unwinding spiral vibrations of rotors are documented in several papers (Black, 1968; Kroon and Williams, 1939; Dimaragonas, 1973; Kellenberger, 1979; Natho and Crenwelge, 1983; Hashemi, 1984; Smalley, 1987; Muszynska, 1993). In addition to the spiral response, Dimaragonas, (1974) described an oscillating mode of shaft vibration, occurring during the transition from the spiraling to a steady-state mode. A similar result from an improved rotor dynamic model was obtained by Muszynska (1993). The problem of rub-related heat distribution was discussed by several authors, for example, Sweets (1966), Kellenberger (1979) and Smalley (1987). The most complete analysis of the heat transfer problem associated with rub is given in the book by Dimaragonas and Paipetis (1983). In all referenced literature the analysis of the shaft bow, resulting from the uneven temperature distribution due to rub, is based on an approximation on the mean flexural rotation of one end of the shaft in relation to the other (Goodier, 1958). The idea of the discontinuous variable transformation applied in this paper for the rub dynamics analytical description appeared first in the paper by Zhuravlev (1978), and was expanded later by Petchenev and Fiddling (1992) and Goldman and Muszynska (1994a,b; 1995a,b). 2. MATHEMATICAL MODEL OF THE RUBBING ROTOR An isotropic rotor in its lateral mode motion is considered (Fig. 1). The rub (or more generally any local nonlinearity which creates a thermal effect) occurs at the shaft axial location 12. If the rotary inertia and shear stresses are neglected, the equations of motion of the rotor can be expressed as follows: m+ 0 mgfZeJ + O, where x(1, t), y(1, t) are horizontal and vertical displacements of the rotor at the axial location IEjO2(7-r),l 0l 2 v/7-1 (1) M2(/’/) Bending Monet to Moment of Inertia Ratio: I(1,) | hermai Bow Mmle L T’ Temlratut difference (1, t) x + jy (x, + jy, " FIGURE Physical model of the rubbing rotor. in the stationary coordinates, ’-x/jy; EJ is bending stiffness, m is mass linear density,/- is a vector ofthermal bending in stationary coordinates, /(l, t) v(1, t)eJ, /, is a vector of thermal bending in rotating coordinates, is rotative speed, -t + Y is a distributed unbalance vector in the coordinate system rotating with the rotor at the particular axial location l, Q- Qx +jQy is a complex vector describing linear density of the external and nonconservative forces. Assuming that the contact/no-contact situation at the axial location 12 generates a radial reaction force distribution over the rotor surface as follows: Of,, off a L/2 -L/2 2r /4/ (...) is defined for angles as a periodic with period of 2. Since the rotor velocity at the contact location can be approximated as R2, the heat rate density g(l, t) per unit area equals to the friction force power per unit area. Taking into account Eq. (4), it can be presented as follows: U- -OKu([T2] c) F2 172] > c o-,IFzl 0 ifl&l < c {1 if x+ (2) (9+-arg(7)). (5) The thermal conductivity equation with initial and boundary conditions where KT is the local stiffness of the stationary obstacle, Y2 7(/2, t), c is the radial clearance between the rotor and the obstacle, the vector Q can be presented in the following form" OZT / 10T 10T O2T Or t=0 + 7 012 at fiT I=0,L =0, 10T -c(1) + q(l)h() + (1 + () +r _x(,e,t), () describes heating of the shaft due to rub. Here T= T(r,,l,t) is a difference of temperatures Here.C(/) is a damping linear density, q(l), ?(1) are radial side-load forces linear density and their angular orientation, respectively, f is a dry friction coefficient, L is the length of the rotor and (... is the Dirac function. The thermal bow appears due to an uneven temperature distribution along the rotor caused by the friction force-generated heat. The latter can be characterized by the heat rate density g(l, 9, t) per unit area ofthe rotor cross section (is an angular coordinate, Fig. 2). Considering the particular area element R2dl d (R2 is the rotor external radius), on the rotor surface around the point with axial coordinate l, the friction force jfN is applied to the rotor ir CN-- deN CN+ deN and -dl +dl (it is assumed that the rotor-to& & stator contact occurs at a single location only), where 9N- 2-- + arg(&) is the angular position of the friction force (Fig. 2). The friction force, therefore, can be expressed in a form of a FIGURE 2 Cross-section of the rotor at the rub axial location between the environment and the material point of the shaft with coordinates 1, r, at an instant t, and ,/ are thermodynamic constants, R1 is the rotor internal radius. To complete the problem formulation, a relationship between the temperature distribution and thermal bow has to be derived. In order to accomplish this, the assumption is made that the rotor can be considered within the limits of Euler’s beam theory. In this case, the thermal stress-related bending moments in the rotating coordinate axes Xr, Yr are as follows: following expression for M(I, 12): n m-Z2l- (Lrn_l,,7 My-- -)T db fR ETrZ R2 cosbdr My(t, 1), Mx ETr 2 sin dr Mx( t, ), Obviously the expression -(M(1,12)/I(12)) determines the mo+de shape of the thermal bow (see Fig. 1) while i?(t) plays the role of the modal time vector-function. The equations which describe the latter can be derived from Eqs. (6) by integral transformations (see Taylor (1924) for details), with the assumption that all thermal processes are limited to the rotor area with the same cross-section defined by internal radius R1(12) and external radius R2(12), and averaged over the yet unknown period of mechanical oscillations (e is a small parameter)" where XT is the thermal expansion coefficient, E is Young’s modulus of elasticity. According to the Castigliano’s theorem and Eqs. (7), the vector fir(l, t) of the thermal bow in the rotating coordinates can be expressed in the form of convolution: dr where \. DT (1 2)[k (1 q- 32) q- /(1 2)] (1-N2)(kl-+-th) q- h lq-t2q-8N3-+-N(1--N2)- where M(1, la) is a bending moment at the axial location la, resulting from the unit load applied at the axial location 1. For simplicity it is assumed that the rotor has constant modulus of elasticity E for all cross-sections along its length. Due to the local character of the rotor heating (Fig. 1) the convolution can be approximated as shown in the second of Eqs. (8). An example of a simply supported rotor with overhung impeller, shown in Fig. 1, has the fR2(12)2 Since, according to Eqs. (8),/3 r is proportional to the temperature ;?- i?i2 di, it follows from the first Eq. (10) that d/3r/dr O(e/3r) has a higher order of smallness than fir itself. As a consequence, the thermal bow in Eq. (1) can be considered as a parameter, and the system dynamics become essentially decoupled from the thermal part of problem. The rotor lateral mode modal coordinates Uq + jVq (q 1,2,..., oo) togetherwithorthogonal modal functions % (q-1,2,..., oo) can be introduced in a way that the following equations are satisfied: as follows: ffq 2,1- c cr- ZAq q=l (1- o-A), Z 7]q(l),q(t) + e Z Z]q(l)(q -4j(p for statically "loose" case (no contact), -1 for statically "tight" case (contact), k=l k=l (l,/2) dl q(l) M1 1(/2) rCq. (12) In this case, Eq. (1) can be written in modal coordinates as follows: where cA is an absolute value of a gap between the rotor static position and the stator. According to physical assumptions, A << 1. The coefficient A is used below as a measure of smallness. (2) One mode in particular, the kth mode, for example, governs the contact at the axial location 12. It means that this mode delivers a much higher contribution into [F(12, t)] than all other modes: o(1), ]q(/2) Here Mq, Dq and Kq are modal mass, damping and stiffness of the qth mode, q mTq dl is a modal unbalance, PqeJTq qeJ/lq dl is a modal radial side-load force. The analysis of Eqs. (13) which represent the mathematical model of the system, is performed below under the following physical assumptions: //q O(x/-) (q In almost all practical situations this is true. (3) The vertical (imaginary) direction for each mode is chosen opposite to the direction of the corresponding radial side-load force: ")/q-3rc/2, q- 1,2, k). New variables are introduced as follows" (1) The system of applied radial side-load forces maintains the static position of the rotor very close to the stationary obstacle at the axial location 12. The rotor-to-stator contact is intermittent due to the dynamic action of the unbalance. (q#k), P/e jzk --jc Kk (1 Ek A/), (14) (q -j /Xq Nq (16) where (u+jv) and rq (qk) are dynamic components of Eqs. (13) solutions, h is the In this case, the absolute value of the shaft static displacement at the rub axial location is P. GOLDMAN et al. nondimensional distance between the rotor and an obstacle at the axial location 12, of smallness, but the conditions (19) of the switch are precise. s- l(12, t)]/(c/ ) 0(1), or- arg((12, t)). 3. MECHANICAL RESONANCES AND With the new set of variables h, u, Iq (q- k) using Eqs. (16) and (13) can be rewritten as follows: HEAT GENERATION The rotor resonance responses to the unbalance which are considered below are associated with the leading mode, or in mathematical terms, are described by the first two equations (17). It means that the rest of the Eqs. (17) determine only "forced" solutions, which are defined at each sequential approximation by the previous approximation to the solution of the first two equations. Therefore, it is important to consider the latter in h" + n(1 -t- @2)h crn AH + O(A2), F utt + nZqFq + o( x :) 2 2 -jOcrnpqh + x/q[aqe j(+cq) -+-7rq, sTeJ(+cv)] + zx + where (17) -2nkh’ sin(g) + c) + sr(er/, + 1) sin(g) + cr) + n2k U- -2nu + eca cos(g) + a) + /r,srcos(g) + at) + Opngh Wq ’q -2gqnqE + Opqnqah u + Dk,q u2 u’2 + Z (n qv/:k n2q)Im r q], (18) U,q 2Mk,qUk,q Kk,q 2 mk,q nk,q U,q Jk akq e,q c2 O(1) O(1) t, "’" gf p,q gf k’q p2 Kk,q -k l’2k-l-/kZ 7rq, 7rq" cA27rq, STejecT, --Trq I(/2) M1 (/2,12). generating approximation, which is defined by neglecting all right-side terms. The second of Eqs. (17) in generating approximation has a simple solution: u= pcos0, p’ =0, 0’= nk, The first of Eqs. (17) in generating approximation, together with the switch conditions (19), is more complex. Its solution can be built using piecewise integration, and connecting The conditions (3) of the contact/no-contact switch now become very simple: 0- f0 ifcrh>0, ifcrh <_ 0. (19) Note, that the right-side terms of Eqs. (17) are calculated with an accuracy up to the second order conditions at the ends of continuity intervals: l+x()p2 l+x()p2 1-5 sin cos (20) where S is constant referred as an amplitude parameter, where Q, F, U, W are functions of new variables, is determined by the system of Eqs. (17) and an integration variable. Equations (22) have the required format, with three fast rotating phases 0, and two slow variables S and p. This allows application of the Averaging Method [7]. Note that at this point, the equations are limited to the terms of the first order of smallness, and only small ., -1 (21) The relation between h and S (amplitude parameter) and (phase) is shown graphically in Fig. 3. It also shows the relation between the amplitude parameter S and overall amplitude A. Equations (20) and (21) constitute a variable transformation. This transformation allows to introduce the rotor vertical response amplitude parameter S and and is used to convert the original "phase" system (17) into the form with three rotating phases 0 and two slow variables S, p: right-sided terms of Eqs. (22) have discontinuities, as they change with the fast rotating phase As it results from the expression (21), the ratio co/nk is contained within the following limits: ,, S’ <--< co -n:-1 + 1/X,/’I _+_p2 if or- +1, ifcr--1. ’ p,_ O, p, ’q(S, )) + O(A2); co coAF(S, p, O, ,’q(S, )) + O(A 2) A U(S, p, O, rq(S, ,)) sin O AQ(S, p, , , ,, , nk + l/V/1 co <--<v/l+ p2 +p2- nk (24) + O(A2); 1-n-0, 1-ico-0 (i- 1,2,3,...), 2n co 0 (for cr -1). 0t- II k zxu(s, p, o, r (s, ))cos0 O(A 2) pnl (22) ’--1 and a number of noncritical variables These inequalities, together with Fourier analysis of the right-side terms of Eqs. (22), show that possible resonances occur when: (25) rq _jcrp2q nq sin fo0 + O(zX) (23) The analysis of Eqs. (22) from the standpoint of balance between the supplied and dissipated energies, allows the stationary resonance solutions for ic 0 (i-- 1,2, 3,... ), or the case of nk 0, for the case of a combinational resonance: n 0 and 2n-co=0. The first resonance occurs when the rotative speed of the shaft f2 is close to the rotor A) Normal-loose case S-I- or-1 B) Normal-tight case contact no contact no contact contact FIGURE 3 Variable nondimensional distance h (c- 721)/c/X as a function of phase leading mode natural frequency vk. It could either be accompanied by the vertical resonance 2nk-a--0 or not. This regime, referred to as a horizontal mode resonance, is described in detail in Taylor (1924). Important thing to know about horizontal mode resonance is that, due to the symmetric heating, the thermal bow does not occur. The sequence of resonances 1-ico(Sa)=O (i= 1,2, 3,... ), referred to as vertical mode resonances, is of interest because it creates uneven heating which results in a thermal bow. The resonance frequency equation according to Eqs. (21) determines the zeroth approximation Sa to the slow variable S as a function of the ratio f/iu in the corresponding resonance zone. A simple analysis of Eqs. (21), together with inequalities (24), shows that for each value of 1,2,... it can be satisfied only within the range of rotative speeds determined by the following inequality: u of the leading kth mode, the subsynchronous 1/2 regime (i 2) occurs at rotative speeds higher then 2u, and so on. In the case of normal-loose situation, (or-+ 1) the maximum rotative frequency of the corresponding resonance regime is lower then 2iu, while in the normal-tight situation 2iuk is a minimal rotative frequency. This agrees with practical observations of rubbing rotor behavior (Choi and Noah, 1987). The parameter p, which affects the width of the frequency band for each regime, characterizes the stiffening effect of the rotor-to-stator contact. Equations (22), after averaging in proximities of vertical resonances (see details in Zhuravlev (1978)), allow for the following stable stationary solutions: + sign(G) arccos bp2 i- 1,2,3,... :r 7rk(1 @/1 n)[(1 +p )nk 1]qkG 1/Sa COSO + ncr/SasinO)’ arccos G(Sa,p) This means that the main, 1 (synchronous) regime of rotor vibrations (i- 1) occurs at rotative speeds higher then unaltered resonance frequency Sa + 7r(1 p2 in: + P)Cr/1 (27) where Sa is a function of the rotative speed, determined by the solution of resonance equation ico(Sa) 0 (i 1,2, 3,...), /- be j kak e j"k + ST(kTrk, + 1)e j"T is an equivalent vector of unbalance which includes a component due to the thermal bow. It defines the series of x, 1/2x, 1/3 x,... regimes in the rotative speed bands roughly described by the inequality (26). The relationship of the vertical response phase ( -t- O/i) and overall amplitude A with rotative speed ratio for the vertical modes, calculated for a particular set of parameters or, p, and b, is presented in Fig. 4. Based on the resonance solutions (27) and expressions (11), the heat generating vector () can be calculated: where Sa is determined by the second of Eqs. (25) and Eqs. (21) as a function of rotative speed f. Since ()) is essentially the forcing function in Eq. (10) for the thermal bow, its behavior determines the behavior of the thermal bow. Figure 5 depicts ()) in a polar plot format for different resonance regimes i= 1,2, 3, 4. ’ 4. SUMMARY AND CLOSING REMARKS This paper outlines the modeling of thermal/ mechanical effects of one of the most destructive malfunctions in rotating machinery: the rotorto-stator rub. The thermal/mechanical problem is partially uncoupled by the assumption that the thermal process is relatively slow. As a result, the rotor thermal bow remains in the mechanical equations as a parameter which can be considered a constant. The combination of the Averaging Method and the assumption that the thermal processes are quasi-static allows the heat transfer problem to reduce to a vectorial ordinary differential equation (Eq. (11)), with the heat generating equivalent vector as a forcing function. This equation allows a realistic estimate of the thermal <) jf fcAKf cr V/Sza + Pz(SZa 1) 27ri(1 e -2jo) / p2) jcos(1 +ngl? +p2) +jcos [1--e-2J9(cos26--jnkV/l+p2sin2(5)]}e j7, [ 00 Rotative speed versus k-th mode natural frequency ratio 3O Rotatlve speed versus k-th mode natural frequency ratio FIGURE 4 Phase and nondimensional amplitude of the rotor-to-stator distance versus rotative speed to natural frequency ratio for the series of Ix, 1/2x, 1/2x, 1/4x resonances. Calculations are made for the parameters cr-- 1, p 2, b/ 4. FIGURE 5 Heat generating vector (} versus rotative speed to the natural frequency ratio in the polar plot format. Heavy pot indicates the position of equivalent unbalance vector b. cr-- 1, p--2, b/ 4. Goldman, P. and Muszynska, A. (1994a), Dynamic effects in mechanical structures with gaps and impacting: order and chaos, Transactions of ASME, Journal of Vibration and Acoustics, 116, 541-547. Goldman, P. and Muszynska, A. (1994b), Resonances in the system of the interacted sources of vibration. Formulation of problem and general results, International Journal for Nonlinear Mechanics, 29(1), 49-63. Goldman, P. and Muszynska, A. (1995a), Rotor-to-stator-rubrelated, thermal/mechanical effects in rotating machinery, Chaos, Solitons, and Fractals, Vol. 5, No. 9. Goldman, P. and Muszynska, A. (1995b), Smoothing technique for rub or looseness-related rotor dynamic problems, Proceedings of 1995 Design Engineering Conferences, Vibration of Nonlinear Random and Time-Varying Systems, DEVol. 84-1. Goodier, J.N. (1958), Formulas for overall thermoelastic deformation, Proc. 3rd Int. Congr. Appl. Mech., John Wiley, New York, p. 343. Hashemi, Y. (1984), Vibration problems with thermally induced distortions in turbine-generators rotors, vibrations in rotating machinery, Third Int. Conf. Proceedings, IMechE, C271/84, York, U.K. Kellenberger, W. (1979), Spiral vibrations due to the seal rings in turbogenerators. Thermally induced interaction between rotor and stator, ASME Paper 79-DET-61, Design Eng. Tech. Conf., St. Louis. Keogh, P.S. and Morton, P.G. (1993), Journal bearing differential heating evaluation with influence on rotor dynamic behavior, Proceedings of the Royal Society, London, A441. Kroon, R.P. and Williams, W.A. (1939), Spiral vibration of rotating machinery, Proc. 5th Int. Congr. Appl. Mech., John Wiley, New York, p. 712. Muszynska, A. (1989), Rotor-to-stationary element rub-related vibration phenomena in rotating machinery, literature survey, The Shock and Vibration Digest, 21(3), 3-11. Muszynska, A., Franklin, W.D. and Hayashida, R.D. (1990), Rotor-to-stator partial rubbing and its effects on rotor dynamic response, The Sixth Workshop on Rotordynamic Instability Problem in High Performance Turbomachinery, NASA CP 3122, College Station, Texas, pp. 345-362. Muszynska, A. (1993), Thermal/mechanical effect of rotor-tostator rubs in rotating machinery, Vibration of Rotating Systems, DE-vol. 60, ASME Design Technical Conf., Albuquerque, New Mexico. Natho, N.S. and Crenwelge, O.E. (1983), Case history of a steam turbine rotordynamic problem: Theoretical versus experimental results, Vibration Institute Proceedings, Machinery Vibration Monitoring & Analysis, pp. 81-89. Newkirk, B.L. (1926), Shaft rubbing, Mech. Eng., 48, 830. Petchenev, A. and Fiddling, A. (1992), Hierarchy of the resonant motions of vibroimpacting system excited by the inertia source with limited power (in Russian), Mechanika Tverdogo Tela, No. 4. Smalley, A.J. (1987), The Dynamic response of rotors to rubs during startup, Rotating Machinery Dynamics, Vol. 2, Eds. A. Muszynska and J.C. Simonis. Sweets, W.J. (1966), Analysis of rotor rubbing, GE Technical Information Series, DF-66-LS-70. Taylor, H.D. (1924), Rubbing shafts above and below resonant speed, GE Technical Information Series. No. 16709. Zhuravlev, B.F. (1978), The equations of motion for the systems with ideal one sided restrictions (in Russian), PMM (Applied Mathematics and Mechanics), Vol. 42, No. 5, pp. 781-788. bending mode (Fig. and Eqs. (9)), and can be applied not only to rub-related heating. The other possible application is an effect of journal bearing differential heating, described in the paper by Keogh and Morton (1993). The heat generating equivalent vector for the rub is calculated by application of the discontinues variable transformation and resonance version of the Averaging Method to the mechanical part of the problem. Based on the heat generating equivalent vector behavior, predictions can be made on the one of three possible thermal bow behaviors: Asymptotic approach to the equilibrium state of the thermal bow. Increasing spiraling motion of the thermal bow in the direction opposite to rotation. Slow oscillations of the thermal bow. The analytical algorithm described in this paper (Eqs. (20), (21), (23) and (27)) has a high potential as a valuable research and prediction tool for investigating rub and thermal effects in rotating machinery.

Journal

International Journal of Rotating MachineryHindawi Publishing Corporation

Published: Mar 9, 2015

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