e s 82-GT-246 Copyright 1982 by ASME

Transcription

1 e s THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Released for general publication upon presentation. Full credit should be given to ASME, the Technical Division, and the author(s). Papers are available from ASME for nine months after the meeting. Printed in USA. Copyright 1982 by ASME 82-GT-246 P. E. Simmons Senior Specialist Engineer, Shell UK Exploration and Production London, England Formerly: Machines Section Engineer, Petrochemicals Division, Imperial Chemicals Industries Ltd. The Response of a Geared Compressor Set to Torsional Excitation Accounting for Damping and Flexibility in the Bearings and Damping and Backlash in the Gear A mathematical model representing the torsional characteristics of a machine train including a gear has been developed incorporating a number of features which are usually neglected, namely, damping and flexibility in the bearings and damping, backlash and pitch error in the gear teeth. This model has been used in conjunction with a computer simulation language to predict the performance of a geared compressor in which there is a torsional resonant frequency close to twice the motor speed. NOMENCLATURE D damping coefficient E half amplitude of pitch error F force G gear tooth pitch error K radius of gyration M mass P gear pitch radius S stiffness coefficient t time x displacement co angular velocity SUBSCRIPTS B backlash (in gear mesh) C compressor D turbine GB gear bearing GT gear teeth M motor P pinion PB pinion bearing R relative (between gear and pinion) 1-10 points on levers representing gear and pinion INTRODUCTION The classic method of torsional analysis of a machine train including a gear is to develop a supposedly equivalent single shaft model. This is done by multiplying all the moments of inertia and stiffness on one side of the gear by a factor equal to the square of the speed ratio. This method neglects the degrees of freedom associated with the ability of the gear shafts to move laterally in their bearings when these are of a hydrodynamic type. It is also incapable of taking account of the damping that will result from any movement in the bearing or from relative movement between the meshing gear teeth (backlash). Such torsional damping that does exist in a single shaft system will normally be very small and is usually neglected. The classical analysis can therefore, only provide confidence in the satisfactory operation of the unit so long as the predicted resonant frequencies do not coincide with a possible exciting frequency. Contributed by the Gas Turbine Division of the ASME.

2 The accuracy of the classical analysis is dependent on the degree of subdivision used in the mathematical model and also on the values assumed for certain empirical factors, for example the penetration factor assumed for shrunk on couplings and other components. Since there is always some uncertainty in the results, it is usual to require that any resonant frequency should be at least 10 percent away from any potential exciting frequency. In the case of the particular machine which was the subject of the study described herein, the manufacturer claimed that it would be satisfactory even though the classical analysis predicted that there was a resonant torsional frequency almost exactly twice the speed of the 4-pole driving motor. There are at least two mechanisms which can potentially generate torsional excitation. Firstly, transients at the instant of starting the motor. These can be very large in amplitude but decay quite rapidly. Secondly, slight ovality in the gear will result in a component of the cumulative pitch error with a twice running speed frequency. The author therefore, decided to attempt to perform a mathematical simulation of the machine train using a model which would allow for the damping in the system neglected in the classical method. This model can then be excited in the manner representing the two mechanisms described above to determine whether the resulting response is acceptable or not. DESCRIPTION OF THE MACHINE TRAIN 1.p SM / The machine D^^ which is the subject of this study I SGT consists of a centrifugal compressor driven through one end by a four-pole induction motor through a step-up gear box and from the other end by a steam turbine See Figure 1. Dp Depending on the process conditions,the power flow may vary from one extreme in which the motor is providing all the power and the turbine just idling; to the other extreme in which the turbine is delivering more power than is required by the compressor and the motor is acting as a generator. The speed of the set can therefore vary between about 1% below to about 2% above the synchronous speed. THE MATHEMATICAL MODEL The mathematical model developed is shown in Figure 2. Angular motions are represented by linear movements at unit radius, moments of inertia are represented by masses and torsional stiffness by linear springs. Fi M@ I( Fo MM I r DM MOTOR KG MG PG r F $ GAR E z ^F3 Fs b&''g 1^5 4D&B XB F 5 ^p B61.0 F PINION F o F8 Ks Kf' F,0 S^ COMPRESSOR r Fc i i DC Motor Gear Compressor Turbine 4-pole Double Induction helical Parallel shaft Sp FD TURBINE MD 1 DD Fig. 1 Arrangement of Compressor Set Fig. 2 The Linear Model The gearbox is of the double helical parallel shaft type. All the couplings are of the flexible diaphragm type and do not, therefore provide any torsional damping. All the gearbox bearings are hydrodynamic oil lubricated; those on the pinion shaft being tilting pad type and those on the gear shaft plain cylindrical. The speed increasing gear is represented by a pair of rigid levers, the larger one represents the gear and the smaller one, the pinion. The pivot point of each lever (points 2 and 9 on Figure 2) is connected to ground through a spring/damper combination 2

3 (SG/DG and Sp/Dp) representing the gear shaft and pinion shaft bearings. The moment of inertia of the gear is represented by two masses (MG) each equal to one half of the mass of the gear and positioned at distance 'Ks' on each side of the pivot point where distance' ng' is equal to the radius of gyration of the gear. The moment of inertia of the pinion is similarly represented by masses 'p' at distance 'KP' from the pivot point. The levers representing the gear and pinion are connected together by a linear link '5-6'. One end (Point 5) is on the lever representing the gear at,v0, from the pivot point; where 'PG' is equal to the pitch circle radius of the gear. The other end (point 6) is on the pinion lever at distance 'PP' from the pivot point; where 'Pp' is the pitch circle radius of the pinion. The link '5-6' which represents the gear mesh incorporates a number of elements:- (i) A linear spring (SGT) representing the flexibility of the teeth. (ii) An element (xb) which represents a dead band in the relationship between load and deflection of the teeth (i.e. backlash). (iii) A damper representing damping due to the presence of oil between the teeth. (iv) An element (E) by which means a sinusoidal disturbance can be fed in representing cumulative pitch error in the gears. The two levers are connected to the other components of the machine train through points 3 and 7 each at unit distance from the corresponding pivot point. For completeness damping elements (DM, DGM, DPB, DC, DD) have been included representing rotational, viscous damping in the bearings. It will be observed that, the gearbox apart, the system has been reduced to a very simple model in which each of the main components, motor, compressor and turbine, is represented by a single mass. A preliminary analysis using the classic method showed that such a reduction did not significantly affect the lower resonant frequencies in particular the third natural frequency which was the one giving cause for concern in comparison with the original model used by the manufacturer which comprised fourteen masses. Obviously additional elements could be added to the model used herein. This would merely require more differential equations to be solved and hence consume more computer time. As it was, one second of simulation time occupired about 15 minutes of actual computer time. The equations which describe the behaviour of this model are listed in Appendix I. THE COMPUTER The computer used in the simulation was a 'Xerox 530' with analogue output facilities and a "hands on" control feature. This latter feature enabled the simulation to be stopped at any stage, the computer interogated to ascertain the value of any parameter and any of the 'constants' modified. The program can then be restarted either from the point when it was stopped or from the beginning. 'Simulation Councils Inc'. It has been used mainly for the simulation of complex process control systems. The language is based on FORTRAN IV and is essentially a centralised integration system facilitating the solution of a number of simultaneous differential equations. It also has a number of standard function generators built into it, in particular a dead band function which was used to simulate backlash in the gear teeth. CONSTANT VALUES The derivation of values of the various constants used is given in Appendix II. The following principles apply in addition to the comments above relating to the gearbox. (i) The masses MM, MC, and MD representing the motor, compressor and turbine respectively are numerically equal to the moments of inertia in units which are consistant with the linear dimensions used in the gearbox model and the units of the springs. (ii) The stiffness of the linear springs SM, SC, and SD are numerically equal to the combined torsional stiffnesses of the corresponding couplings and shaft-ends also in consistant units. (iii) The damping coefficients DM, DGB, DpB, DC, and DD, have been estimated from the known bearing geometry and an assumed oil viscosity. EXCITATION The model has been designed so that it can be excited in the following two ways. Cumulative Pitch Error A facility has been provided to apply a sinusoidal disturbance in the relative positions of the teeth on the gear and the pinion, Both the amplitude and frequency of this disturbance can be varied. The cumulative pitch error of the gear and the pinion can be resolved into a series of sinusoidal components at different frequencies. The component of prime concern is this particular study being the 2nd harmonic of the gear rotational speed. Such a component would, for example result from slight ovality in the gear. Oscillating Motor Torque When an induction motor is started by instantaneous connection to the mains supply the air gap torque will initially have a large oscillating component at supply frequency (see reference 1). The amplitude of this oscillating torque can be several times greater than the full load torque. However, generally it will decay away rapidly over a period of less than one second. The air gap torque on the motor represented in the model by force FO (see Figure 2) is modelled as sinusoidal function in which the amplitude is also a function of time. The actual form used is:- F0 = 2 F I 1 + sin f fl (2V/t - 2) } l sin (cot) THE PROGRAM The programme was written using a commercially available simulation language SL-1 developed by The reason for this particular form is explained in Appendix II 3

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5 (c) To determine the effect on this response to gearbox excitation of the load distribution between the motor and turbine. When the turbine is carrying most of the load, the load transmitted through the gear is small and therefore there is a greater tendency for tooth separation to occur. (d) To simulate the starting of the motor and the transient oscillating torque occurring at that time and determine if this could result in unacceptable oscillating torque. In this case the supply frequency was assumed to be the resonant frequency determined above. To satisfy these objectives the following procedure was adopted. (i) The model was excited at the gearbox with an amplitude of 0.001" at the gear mesh. The frequency of this excitation was varied around the expected resonant frequency of about 60 Hz. In each case the simulation was allowed to run until equilibrium was reached i.e. the oscillations in all parts of the model were no longer increasing. The amplitude of the maximum oscillating torque was plotted against frequency and hence the resonant frequency and the response to excitation at this frequency were found. (ii) Exciting the model at the resonant frequency determined as above the constant load in gearbox and hence the force holding the teeth in contact, was varied from full load down to zero to determine at what point tooth separation occurred and the consequencies thereof. (iii) The model was excited at the motor at the resonant frequency using the forcing function given in paragraph above and described in more detail in Appendix III. The transient oscillating torque was assumed to reach a maximum value of about four times full load torque. DISCUSSION OF RESULTS The results of a typical run in accordance with the procedure outlined in (i) above are shown in Figure 3. It can be seen that the maximum amplitude occurs at the compressor with the turbine moving in antiphase to it and very little movement in the rest of the system. This is in agreement with the modal shape determined by the manufacturer using the classical method. This modal shape is shown in Figure 4. The maximum oscillating torque occurs at the coupling between the compressor and turbine and in this particular case reaches a valve of 4000 lb ft with an excitation frequency of 3650 per min, and equilibrium is reached after about 1.2 secs. In Figure 5 the amplitude of the maximum torque determined as above is plotted against excitation frequency. It can be seen that the resonant frequency is about 3625 per min and the maximum amplitude is about 7000 lb ft. The steady state full load torque is also shown on this figure. Because, in this particular mode, the amplitude of the oscillating load on the gearbox is relatively w Z o -4 -i a cq E 0 a p w p o o U ts U u u Fig. 4 Model Shape as Predicted by the Classic Method ti U. 6 O 4 W t z 7 350o FPGQuENCY cpnn. Fig. 5 Maximum Oscillating Torque at Coupling between Compressor and Turbine small, tooth separation does not occur until the steady state load on the gear is reduced to a low value i.e. the turbine is carrying most of the load. A typical result is shown in Figure 6 in which the gearbox steady state load is about 5% of the full load. The point at which the teeth separate can clearly be seen. However the resulting torques are less than without tooth separation assumedly due to the additional damping occurring and the fact that when the teeth are not in contact the resonant frequencies will be different since the two halves of the machine train are uncoupled. For a similar reason when the load on the gear was reduced to zero, the system did not respond at all to excitation at this frequency. Figure 7 shows the effect of exciting the system at the motor using the forcing function described above and in Appendix III. The way in which the energy in the resulting oscillation feeds through from the motor to the compressor and turbine can clearly be seen. It can also be seen that because the excitation occurs for only a limited period the resulting oscillating torques are of lesser amplitude than when the system is continuously excited from the gearbox. z 5

6 10 Ft'.lb. o se:^ xms-uunr : s s es. a en :: p!i^ -!-!-,y '.r..e......r eiee iis tst stutu::us;:$^i-ns-she. Hs,HIH:ulm-alli:ulluil^ iiinlliii iei^ -seiiiijim ise3s:s::::l::e:1e::ti:::tnttiestg ss =ssst:at:s e5eee=ee:s :::ss:::e9ee5=e U _^ D z Sxld^ 3 ^+ l z O -SXIo 3 I oz LL ^.s Z I0z 0) z O 0 k- _ I pz 'o 2 - l oz LJLJI If_1- Lam_ 1- LL + ± - +-I I---±o o 3 0 0S. TIME Secs. 06 o Fig. 7 Response to Transient Motor Starting Torque

7 Z o_ I- I'-) Li U- Iu J Q z o_ O Ft ill I, $ 1^11I Ic tvr-, -r-t-i _/ j LiLf z I I F ^^- I fd rt-rt T-T - T - I Qz L_I-_I 1-1- _ 10 r- - LT _Lr- I I F I i I r CONCLUSIONS A mathematical model representing the torsional characteristics of a machine train including a gear has been successfully developed incorporating the following features which are usually neglected. (a) Flexibility and damping in the gearbox bearings. (b) Flexibility, damping and backlash in the gear teeth. (c) Rotational viscous damping in the (d) bearings. Cumulative pitch error in the gear teeth. This model can be used to determine the response of the system to torsional excitation at resonance. In the case of this particular machine train studied, a 'twice-per-rev' component of the cumulative pitch error of the gear of amplitude of inches will result in an oscillating torque at the coupling between compressor and turbine of amplitude of 1.4 times full load torque. When all, or nearly all, the load is being carried by the turbine rather than the motor, the maximum oscillating torque is lower than reported in the previous paragraph because tooth separation occurs and this "de-tunes" the system. The system will also be excited by the transient oscillating torque produced by the motor at "switchon". However, at the expected level of amplitude and duration of this excitation, the resulting torques are less than those resulting from gear tooth excitation. The maximum amplitude of the oscillating torque being about one half of the full load torque. ACKNOWLEDGEMENTS The author is indebted to Mr P.B. Cook for his assistance in developing the program and for his and Mr B. Hamshere's help in "driving" the computer. The author is also indebted to the director of the Petrochemicals Division of ICI for permission to publish this paper. REFERENCE Peeken, H, Troeder Chr and Diekmans, G "Technical Vibration during the Starting Process in Driving Systems with Three Phase Motors", Second International Conference, Vibrations in Rotating Machinery, Cambridge England 2-4 September Zx)v Ib, 0 0a 0 z TIME - secs. Fig. 6 Response to Gear Excitation at Resonance with Tooth Separation APPENDIX I Equations representing the system. Driving functions. G = E sin (cot) (1) F0 = 2F [ 1 + sin{ 7T(2q/t - 2)}7 sin ([it) (2) Equations of motion MM X N + DM X M = FO - FM (3) MG X1 = -F 1 (4) MG X 4 '_ -F (5) Mp X8 = F8 (6) MP x 10 = F10 (7) MCX D + DC Z C = FC - FD (8) 7

8 MD JC D + DD JC D = FC Equations of geometry X2 = 2 ( )C1 +X2 ) (10) KG XG = 2 C4 -x) (11) x 3 x2 + XG (12) X 5 = ^C2 + PG X G (13) X 9 = z (X 8 + ) 10 ) (14) KP X p = z (X 8 -X0) (15) X 7 =X9 + xp (16) (9) X5 = x9 + P X P (17) XR =X5- X6 + G (18) Force balance equations FM = SM ( D M -x3 ) (19) F 2 = SG X 2 + D G x 2 (20) F 3 DGB X 3 (21) F 6 = F (22) F7 =DPB X 7 (23) F9 =-S )c 9- D X 9 (2k) F C Sc ( x7 - XC ) (25) F D = S D (X - X0 ) F1 = z (F 2 +F 3 +F5 -FM - (F 3 +P GF5 F 4 = z(f 2 +F 3 +F 5 - FM + (F 3 +PGF5 F8 = (F6 + F 7 + F 9 - F G + (PPF6 + F 7 F 10 = 21F 6 + F 7 + F9 - FC - (PPF6 + F7 X R <- X B. F5 S GT + ) B ) - FT + - XB _< x R G 0 F5 = -FT + DGT 'X" R (26) - FM )/KGI.(27) FM )/KG I (28) FC )/Kp} (29) - F C )/Kp}(30) SGT xr (31) (32) JCR > 0 : F5 = SGT xr - FT + DOT X R (33) APPENDIX II Masses Numerical values of constants. Motor MM = slugs at 1 ft radius (1308 kg at 1m radius) Compressor M C = slugs at 1 ft radius (20.78 kg at 1m radius) Turbine MD = slugs at 1 ft radius (32.0 kg at 1m radius) Gear MG = slugs (7.73 kg) Pinion MP = 9.23 slugs (0.63 kg) Stiffnesses Motor - Gear SM = 1.75 x 106 lb/ft (25.5 x 10 N/m) Pinion - Compressor S = 2.0 x 10 6 lb/ft 6 (29.2 x 10 N/rn) Compressor - Turbine SD = 3.07 x 10 lb/ft 6 (44.8 x 10 N/m) Gear Shaft Brgs S G = x 106 lb/ft 6 ( x 10 N/m) Pinion Shaft Brgs S = x 10 6 lb/ft ( x 10 6 N/m) Gear Teeth SGT = 770 x 106 lb/ft (11.2 x 10 9 N/m) Damping Coefficients Motor Shaft DM 0.1 lb sec/ft (1.46 N. sec/m) Gear Shaft D GB 0.16 lb sec/ft (2.33 N. sec/m) Pinion Shaft DPB lb sec/ft (0.182 N. sec/m) Compressor Shaft D C lb sec/ft (0.7 N. sec/m) Turbine Shaft DD lb sec/ft (0.452 N. sec/m) Gear Bearings D G x 10 lb/sec ft ( x 10 6 N. sec/m) Pinion Gearings D x 106 lb sec/ft ( x 10 6 N. sec/m) Gear Teeth DOT 1.0 x 10 3 lb sec/ft (14.6 N. sec/m) Dimensions of Gears Gear rad. of gyration KG ft (0.37 m) Pinionrad. of gyration K ft (0.078 m) Gear pitch radius P G ft ( m) Pinion pitch radius P ft ( m) APPENDIX III Simulation of Motor Transient Starting Torque In reference 1 it is shown that at the instant of "switch-on" an induction motor produces an oscillating torque at supply frequency. The magnitude of this oscillating torque can be as high as four times the motor full load torque. However, it decays away quite rapidly - in less than one second. 8

9 To simulate this transient effect, initially an equation of the following form was used F0 = F sin Cc.) e -Bt where B represents the time constant of the decay of the transient torque However, with this expression the amplitude is a maximum at t = 0. This results in an impulsive "kick" to the system which causes all the displacements to drift-off scale. This is inconvenient when an analogue output is being used. The form of expression eventually adopted i.e. F0 = zf L II + sin f lj(2 Vt - 2 ) } J sin (tot) largely avoids this problem since at t = 0 the expression L II + sin f TT (2t - 2) } also equals zero. 9