A new field balancing method of rotor systems based on holospectrum and genetic algorithm

Applied Soft Computing 8 (2008) 446–455 www.elsevier.com/locate/asoc

A new field balancing method of rotor systems based on holospectrum and genetic algorithm Shi Liu b,*, Liangsheng Qu a a

Research Institute of Diagnostics and Cybernetics, Xi’an Jiaotong University, Xi’an 710049, China b Guangdong Power Test & Research Institute, Guangdong 510600, China Received 7 March 2005; received in revised form 6 November 2006; accepted 22 November 2006 Available online 18 March 2007

Abstract Field balancing of flexible rotor system is a key technique to reduce turbine vibration in power plants. Traditional balancing methods are generally based on the information from a unidirectional sensor. In fact, the motion of a rotor system is a complex spatial motion, which cannot be objectively and reliably described with just a unidirectional sensor in one bearing section. In order to give an accurate description of rotor vibration responses, multi-sensor fusion instead of a single sensor should be used with the purpose of more comprehensive utilization of information. Based on this theory, the paper presents a new balancing method for rotor systems named the holo-balancing method, which successfully applies the holospectral principle in traditional balancing methods of flexible rotor systems. At the same time, genetic algorithm (GA) optimization and computer simulation are used to simplify balancing process. The new method decreases test number, increases precision and efficiency of field balancing. The principle and detailed procedures of the new method are explained in this paper, and the effectiveness of the new method was validated by field balancing of several 300 MW turbo-generator units. # 2007 Published by Elsevier B.V. Keywords: Field balancing; Holospectrum; Flexible rotor; Genetic algorithm

1. Introduction Information science is the most active one in the contemporary era, which developed quickly, and has been seeping through various domains of the national economy. In machine industry, information technology is adopted to improve and expand the function of mechanical products, fully utilize various dynamic information of operating equipments and therefore comprehensively promote the level of manufacturing industry. Information fusion is one of the key techniques informationalizing mechanical products. A large amount of statistical data indicates that the mass unbalance of the rotors is usually the major cause of excessive vibrations of large capacity turbine. In these huge power plants, a rotor is always consisted of several flexible shafts, which are coupled together by rigid or flexible couplings. Although the shafts would all have been balanced at a high speed in manufactory, after assembled together on-site, their balance conditions change due to coupling-affect between each of them.

* Corresponding author. Tel.: +86 20 851 23467; fax: +86 20 87318130. 1568-4946/$ – see front matter # 2007 Published by Elsevier B.V. doi:10.1016/j.asoc.2006.11.012

The field balancing is an ordinary way to reduce shaft vibrations to a given level. In recent decades, the balancing theory has been thoroughly studied, and various balancing techniques have been developed. However, the existing balancing methods are still potential for improvements in accuracy and efficiency. Firstly, nowadays, all general balancing methods (either modal balancing or influence coefficient method) need large numbers of trial runs to obtain the vibration responses of trial weights in different correcting planes. On the other hand, the vibration measurement always depends on a single sensor in one measuring section. This measurement method is based on the assumption that a rotorbearing system has an equal rigidity in different directions, so considerable errors would occur if obvious difference were found among rigidities. In this case, waveforms in different measuring directions will result in different FFT magnitude spectra, different correction masses, and even contrary balancing projects. The motion of a rotor system is a complex spatial motion, which cannot be objectively and reliably detected with just one single sensor. The multi-sensors fusion is an effective technique to describe the rotor’s spatial motion. Nowadays, to obtain the

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comprehensive vibration information of a rotor, two mutually perpendicular proximity transducers are mounted in each bearing section to monitor the vibration in large-capacity steam turbo-generator sets, allowing application of the holospectral technique based on the information fusion. The two-dimensional holospectrum is constructed from two FFT spectra of the vibration signals in two mutually perpendicular directions of a bearing section, which synthetically utilizes frequency, amplitude and phase information. If corresponding frequency components in two spectra are combined properly, a series of ellipses, circles or simply straight lines can be obtained. Based on the two-dimensional holospectrum, the three-dimensional holospectrum integrates full information of rotor vibration simultaneously in all bearing sections [1,2]. Since the frequency, amplitude and phase information is fully utilized, the holospectral technique based on the multi-sensors fusion can improve the balancing accuracy and efficiency [3]. In this paper, the holospectral technique based on conventional FFT spectra and information fusion is applied in the field balancing of a flexible rotor system. A new balancing method for rotor systems named as holo-balancing method is presented. Furthermore, the computer simulation and genetic algorithms (GA) are adopted in this method, which is helpful to simplify the balancing procedure and enhance the balancing accuracy and efficiency. 2. Technical key points of holo-balancing 2.1. Holospectral technique Fig. 1 shows the arrangement of sensors in a 300 MW turbogenerator unit. The rotor vibration can be sensed by two eddy current probes perpendicularly mounted across each bearing section. This paper defines that the phase is the lag angle of keyphasor pulse signal, which is relative to the first forward zero of the vibration waveform. The vibration components with rotating frequency in the ith measuring section, derived from two mutually perpendicular directions X and Y, can be expressed as 

xi ¼ Ai sinðvt þ ai Þ ¼ sxi sinðvtÞ þ cxi cosðvtÞ yi ¼ Bi sinðvt þ bi Þ ¼ syi sinðvtÞ þ cyi cosðvtÞ

(1)

where Ai, Bi are amplitudes; aI, bi the initial phases; v the rotary speed (rad/s); sxi and cxi the sine and cosine coefficients of the signal xi; while syi and cyi are the sine and cosine

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coefficients of the signal yi. In general, the orbit constructed by the signals xi and yi is not a circle, but an ellipse. Thus, the motion of a rotor is a complex spatial motion, which cannot be objectively and reliably detected with just one single sensor. In most traditional balancing methods, only the vibration information in one measuring direction is used, which is based on the assumption of equal rigidity in different circumferential directions of rotor-bearing system. The analysis errors would occur when the rigidity is different. From the viewpoint of information fusion, the holospectrum integrates the two vibration signals in a bearing section as a whole but not as individual measuring directions, which can fully reflect the vibration behavior of a rotor system. The initial phase point (IPP) is defined as the point on 1X ellipse (rotating frequency orbit), where the key slot on the rotor locates straightly opposite to the key-phasor. In this paper, the IPPs are used to analysis vibrations of a rotor system in balancing process, and may reduce errors and improve the balancing accuracy. For the convenience of vector processing in balancing, the first harmonic frequency ellipse can be expressed as an array ri ¼ b sxi ; cxi ; syi ; cyi c ;

i ¼ 1; 2; . . . ; n

(2)

where n is the number of measuring sections. The coordinates of IPPs in first frequency ellipses are IPPi ¼ b cxi ; cyi c . The three-dimensional holospectrum integrates all first frequency ellipses, and therefore provide full vibration information of a rotor system as a whole simultaneously in all bearing sections, which can be expressed by the following matrix 2

3 2 sx1 r1 6 r2 7 6 sx2 6 7 6 R ¼ 6 .. 7 ¼ 6 . 4 . 5 4 .. rn

sxn

cx1 cx2 .. .

sy1 sy2 .. .

3 cy1 cy2 7 7 .. 7 . 5

cxn

syn

cyn

(3)

In this paper, the matrix of 3D-holospectrum is used to describe comprehensive vibration responses of a rotor system. Under the precondition of the linearity, it can simplify the computing of the vibration responses. For example, the initial unbalance responses are 

xi ¼ Ai sinðvt þ ai Þ yi ¼ Bi sinðvt þ bi Þ

or

Fig. 1. The arrangement of sensors in a 300 MW turbine generator unit.

ri ¼ b sxi ; cxi ; syi ; cyi c ;

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and added the trial weights, the vibration responses with trial weights are  0 xi ¼ A0i sinðvt þ a0i Þ or r0i ¼ ½s0x j ; c0x j ; s0y j ; c0y j ; y0i ¼ B0i sinðvt þ b0i Þ then the net response caused only by trial weights in the ith measuring section can be expressed as Dri ¼ r0i  ri . For the vibration responses in all measuring sections, the matrix formula is DR ¼ R0  R

(4)

2.2. Transfer matrix For the machine set which has m balancing planes A, B, and C, . . ., M and n measuring planes, the following matrix displays the 3D-holospetrum representing the unbalance responses caused only by a unit trial weight 1000gn08 on plane A. 2 3 r1 6 r2 7 6 7 RA ¼ 6 .. 7 (5) 4 . 5 rn n4 Similarly, unbalance responses due to unit trial weight on other balancing planes can be, respectively, expressed as RB, RC, . . ., RM in a matrix format, namely the transfer matrix of a unit trial weight. Different from traditional influence coefficient aij, which only represents vibration responses caused by a unit trial weight on plane i in a certain single direction X or Y of the jth measuring plane, the transfer matrix including much more information. The transfer matrix fully utilizes the information from all sensors in a rotor system, represents the synthetical influence of vibrations in all bearing sections due to unit trial weight 1000gn08 added on a certain balancing plane. The transfer matrix is the basis of holo-balancing for a flexible rotor system. The transfer matrix can be obtained through tracking one field balancing, or by a trial weight adding experiment, or theoretical calculation. For a new unit, its transfer matrix may be modified based on a given transfer matrix of its type, according to actual balancing result.

It is the precondition of linearity hypothesis that unbalance is the major fault; meanwhile the rotary speed, load and lubrication action should all keep stable. Accordingly the vibration responses due to all correction weights may be obtained by means of simple addition or subtraction procedures on the grounds of the transfer matrix. In a similar way, if there are serval unbalance masses distributing on the rotor system, the total balancing can be achieved if they are cancelled, respectively. The holobalancing method for flexible rotor systems is based on the prerequisite of system linearity. In practice, the results of field balancing have always confirmed the linearity hypothesis as expected, even if the capacity of a balanced turbo-generator unit is more than 300 MW. The following example comes from certain field balancing. Firstly, the transfer matrices of balancing planes have been obtained with the field experiment. After a group of appropriate correction weights are added on rotor sets, vibration responses are measured and a three-dimensional holospectrum is shown in Fig. 2b. Through computer simulation, we can apply the predetermined transfer matrices and the original vibration matrix in the simulation of field balancing result. Fig. 2a shows the simulated residual vibration, which is substantially consistent with field-measured results in shapes and sizes of ellipses and the phases of IPPs. This case provides supporting evidence for the hypothesis that if the rotary speed, load and lubrication action are all keep stable, it will promise accuracy in field balancing. In the traditional influence coefficient method, as well as the above linearity hypothesis, it is also hypothesized that the vibration response of a rotor from a single sensor is in direct ratio to the unbalance vector, which will lead to a biggish analysis error when the rigidities of a rotor-bearing system in different directions diverge. In holo-balancing method, angle compensation is used to correct the errors due to different rigidities. 2.4. Angle compensation

(6)

The revolution of the rotor in space tracks an ellipse orbit around a geographical center. Although bearing the same period as the synchronous axial rotation of a rotor, its revolution angle speed near the major axis of ellipse is slower than that near the minor axis. Only if the rotor moves from the major axis endpoint to the minor axis endpoint, the angle of revolution is the same as that of rotation. Thus, in general, the whirling angle d of initial phase point (IPP) in revolution orbit (rotating frequency or 1X ellipse) does not equal to the angle of rotation u in the round, i.e. d 6¼ u. Moreover, the larger the eccentricity of rotating frequency ellipse is, the greater the difference between d and u is. As shown in Fig. 3, when the initial phase point (IPP) moves from point x1 to x2, the relationship between d and u can be illuminated as ( na o na o u ¼ arctan tanðb  ’Þ  arctan tanða  ’Þ b b d ¼ ðb  ’Þ  ða  ’Þ (7)

where f(*) represents vibration responses; QA, QB, QC, . . . represent the trial weights on different balancing planes.

where w is the dip angle of major axis of rotating frequency ellipse, a and b are respectively the 1/2 length of major and

2.3. Linearity hypothesis The linearity hypothesis is defined as: the total vibration response due to influence of all trial weights on different balancing planes equals to the sum of individual vibration responses caused by each trial weight. For a linear rotor system, relationship should exist as follows: f ðQA ; QB ; QC ; . . .Þ ¼ f ðQA ; 0; 0; . . .Þ þ f ð0; QB ; 0; . . .Þ þ f ð0; 0; QC ; 0; . . .Þ þ   

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Fig. 2. Verified linearity hypothesis.

minor axis. In the holo-balancing method, when the position of unbalance moves a certain angle, the IPP in 1X ellipse shall move along. The relationship between the two angles is equivalent to that between the rotation angle u and whirling angle d, under the condition that all parameters of a rotorbearing system are kept stable. Thus, if we wish to find the position of unbalance through above relationships, i.e. calculate the angle of rotation by the whirling angle, the angle compensation must be taken into consideration. Based on the hypothesis that a rotor-bearing system has an equal rigidity in different directions, the traditional influence coefficient method only utilizes the information from a single sensor in balancing calculation. It is considered the whirling orbit is a circle, namely the angle of rotation is equal to the whirling angle in a unit time. The angle computation error must be brought into balancing calculation, which will lead the influence matrix to be a singular matrix. The accuracy of balancing shall be decreased.

loaded with trial weights remains in the original ellipse, i.e. trial weights only results in the movement of the IPP in the original ellipse, without any change of the ellipse in shape and size, by connecting this type of initial phase points due to pure trial weights, such as the point Q, a phase shift ellipse could be developed, which is same as the original 1X ellipse in shape and size but with a shifted center. To be brief, the phase shift ellipse also is an ellipse with its centre O0 located as the mirror image of IPP in the original 1X ellipse, shown by dashed line in Fig. 4a. Fig. 4b clearly shows that the final vibration response will be decreased if the IPP of trial weight ellipse Q locates in the shaded area in the phase shift and the original ellipses. The total vibration response will increase if the IPP of the trial weight ellipse Q0 locates outside the shaded area. Under the ideal conditions, total vibration response can be reduced to zero if the IPP of trial weight ellipse Q is the mirror image O0 of IPP of original 1X ellipse. 2.6. Defining unbalance as the leading fault

2.5. Phase shift ellipse If the initial phase point of the original 1X ellipse in one bearing section moves from point IPP to IPP0 after the trial weights are loaded, the vector OQ indicates the vibration response only caused by trial weights. If the initial phase point

Fig. 3. Angle compensation.

Although rotor unbalance is the main cause of rotating frequency faults, the field high-speed balancing is not the general way to solve vibration problems due to various faults. Besides rotor unbalance, the typical faults with rotating frequency include temporary hot bending, permanent bending, change of bearing rigidity, instability of oil film, oversize bearing gap, change of bearing elevation, etc. In order to distinguish rotor unbalance from other rotating frequency faults and identify the type of rotor unbalance, the holospectral techniques are applied, for example, the tend analysis of initial phase point. If the unbalance magnitude is changed, while the phase of unbalance remains unchanged, the position of IPP will remain in the same azimuth. If the unbalance magnitude remains unchanged, IPP will move along with the phase change of unbalance. Fig. 5 shows the 1X ellipses of two bearing sections in one turbine at loading stage, which is a temporary hot bending fault. The root of the blades in the machine was

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Fig. 4. Phase shift ellipse.

cooled by the exhausted gas, which was reintroduced into the cooling holes uniformly arranged in the circumference of the rotor. If some of these holes were blocked, the temperature differential in the rotor would cause a more significant temporary hot bending. An extraordinary vibration was developed as the gradual expanding 1X ellipses in Fig. 5, while the butterfly valve would open by a little percentage and the exhausted gas begin circulating through the cooling holes in loading stage. However, the phases of IPPs on 1X ellipses nearly remained unchanged, and the tangent lines on IPPs remained parallel, which shows that rather than unbalance, the fault is caused by thermal bending [1,4]. This practical case study shows that prior to field balancing, the holospectral techniques mentioned above are effective in recognition and diagnosis of vibration faults. The identification of the leading fault with the holospectrum is the sole pre-requisite of successful balancing of large rotating machinery. 2.7. Optimization of correction weights Based on the general principle of system theory, theoretically speaking the unbalance response on each measuring plane can be cancelled by mounting correction weights on the most relevant balancing plane. However, coupling effect between different balancing planes will influence balancing accuracy. In order to

get optimal balancing results, the genetic algorithm (GA) and computer simulation have been successfully applied [5–7]. In this paper, the genetic algorithm (GA) will be used to optimize the correction masses for a better vibration distribution on the entire rotor system. Implementation of GA is as follows: 1. Utilize transfer matrices to get correction masses and angles on each balancing plane. 2. The angles of balancing weights still stand; apply appropriate correction masses to realize optimization. Firstly, the magnitude of the mass on each balancing plane is encoded by gene. The length of each gene is N, which can be obtained by the equation   log10 ððPmax  Pmin þ DÞ=DÞ N ¼ Round þ1 (8) log10 2 where Pmax is the upper limit of the correction mass, Pmin the lower limit of the correction mass, D the interval mass, Round(*) represents rounding down or up to the nearest integer. Then, the total length of chromosomes is L = M  N, where M is the number of balancing planes. 3. Initialize parameters of GA, including initial population size 400, crossover probability 0.65, mutation probability 0.05, maximum generation 20 or other termination conditions.

Fig. 5. 1X ellipses of two bearing sections in one industrial turbine at loading stage.

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4. Taking the minimal mean vibration on all bearing sections and the best vibration distribution as the aim of optimization, the fitness function is defined as F ¼ MeanðV i Þ  fMaxðV i Þ  MinðV i Þg

(9)

where Mean(Vi) is the mean of vibrations on all bearings, Max(Vi) the maximum value of vibrations, and Min(Vi) is the minimum value of vibrations. 5. Perform GA on each individual, including fitness calculation, individuals selection based on roulette wheel selection, and generation of new offspring through crossover and mutation. Ultimately, an optimal solution will be developed, with the achievement of the termination conditions or maximum generation. Nowadays, efficiency is one of the key points in field balancing, which requires a better balancing result with fewer balancing planes and less test runs. It is problematic for balancing calculation that a comparatively small number of balancing planes would lead to inadequacy of calculating condition. Thus, as well as GA used to optimize the balancing results, computer simulation software has been introduced to imitate the balancing process, which can display the real-time variation of balancing results with three-dimensional holospectrum when the magnitude and azimuth of correction masses change. A better balancing scheme could be chosen from the simulation results, avoiding repeated test runs of a turbine generator set and therefore increasing the balancing efficiency. 3. Procedure of holo-balancing Firstly, two proximity transducers have been mounted adjacent to each other perpendicularly in bearing section, monitoring the vibration with key-phasor. 1. Track a complete field balancing process of a given rotor set, from test runs with loaded or unloaded trial weights and determine the transfer matrices of three-dimensional holospectra RA, RB, RC, . . ., RM, which are the characteristic parameters of rotor systems, providing basis for further holobalancing. The standard form of transfer matrices should be transformed to those with unit trial weight 1000gn08. 2. Utilize the holospectral techniques, before further balancing operation; make sure that the leading fault is mass unbalance. 3. Measure the original vibration of each bearing section in no load trial running; pre-process vibration signals with

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key-phase signal and structure the three-dimensional holospectrum of the initial unbalance response R0. 4. Focus on one bearing section, the magnitude and azimuth of correction mass on the most relevant balancing plane can be determined, following the rule that the IPP of a trial weight ellipse could be considered as the mirror image pair of the IPP of original vibration ellipse. 5. The genetic algorithm (GA) and the computer simulation are used to optimize the correction masses for a better vibration distribution on the entire rotor system. The whole process may be simulated with computer. 6. Add the correction masses to complete the balancing operation, and then modify the transfer matrices complying with the balancing results. 4. Field balancing cases 4.1. Single plane balancing The case is a single plane balancing of a 300 MW turbogenerator unit, in which the relative shaft vibration in bearing No. 1 is usually up to 270 mm. Since the vibrations in other bearing sections are within the alarm threshold value, balancing was performed only on the shaft of high-pressure cylinder, with purpose to reduce the vibration of bearing No. 1. Table 1 displays all data in balancing process, which were sampled from two perpendicular transducers in one measuring plane, with the amplitude given in mm and the phase angle in degree. The three-dimensional holospectrum of original vibrations is described as R0 and shown in Fig. 6a. After estimation of the azimuth of unbalance mass from the phase of vibration in bearing No. 1, as well as the mechanical lag angle, trial weights QA = 610gn2508 was installed in the front bearing box (defined as plane A). Then turbine started up to the rated speed, and the vibration data of the rotor system with trial weights were measured as illuminated by the three-dimensional holospectrum R1 in Fig. 6b. Through previously mentioned vector operation, unbalance responses due to pure trial weight on balancing plane A can be derived by subtracting of the holospectrum R1 with trial weight to the original one R0, also demonstrated by the 3Dholospectrum in Fig. 6c. Analysis of the original vibrations in Fig. 6a shows that the coupling unbalance, characterized by the intersecting generating lines between bearing No. 1 and No. 2, is the major unbalance component of shaft in the highpressure cylinder. Meanwhile, the intersecting generating lines between bearing No. 1 and No. 2 in Fig. 6c show that the

Table 1 Balancing data (mmn8) 1X component

Original vibrations With trial weights Residual vibrations

Bearing No. 1

Bearing No. 2

Bearing No. 3

Bearing No. 4

1X

1Y

2X

2Y

3X

3Y

4X

4Y

251n318 173n282 73n328

132n49 85n24 59n66

134n158 89n92 39n268

81n269 37n200 33n345

87n247 25n236 119n281

75n13 43n55 93n24

71n33 53n348 42n91

42n122 47n85 21n177

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Fig. 6. Calculating transfer matrix: (a) original vibrations; (b) vibrations with trial weight QA; (c) vibrations due to pure trial weight QA; (d) the transfer matrix in plane A. The thin lines represent the original vibrations, and the bold ones represent the simulated results.

unbalance response due to pure trial weight added in plane A could be explained as coupling unbalance. Thus, the indication could be that appropriate correction mass, although loaded only in the front bearing box, can simultaneously cancel vibrations in both No. 1 and No. 2 bearings. According to above discussed methods, including linear hypothesis and angle compensation, we obtained transfer matrix RA representing unbalance responses caused only by unit trial weight 1000gn08 added on A plane in the front bearing box, as demonstrated by the 3D-holospectrum in Fig. 4d. 2

231:2 6 165:0 RA ¼ 6 4 2:3 16:0

86:4 127:3 103:2 80:8

22:2 89:7 85:0 34:2

3 105:7 86:6 7 7 0:5 5 32:5

The correction mass was calculated complying with the phase shift ellipse rules that total vibration response can be

decreased to zero if the IPP of a trial weight ellipse is the mirror image of an original unbalance response IPP. Calculation of correction mass that should be added on plane A is PA = 1082.0gn2078. Aided by computer simulation software, balancing result with correction mass PA was simulated as shown in Fig. 7a. Then, through computerized microadjustment, the optimal result could be developed, suggesting that the actual correction mass should be modified to P0A ¼ 802:0gff217 . As can be seen from Fig. 7b, the simulated balancing result with the correction mass P0A is better than that in Fig. 7a. After actual load of the correction mass P0A on plane A, the residual vibrations were measured, and its characteristic parameters were listed in Table 1. The 3D-holospectrum of residual vibrations was shown in Fig. 8. After field balancing, axial vibrations in both No. 1 and No. 1 bearings were reduced by more than 90%, and vibration in bearing No. 3 increased slightly within the alarm threshold value. Comparing Fig. 8 with Fig. 9, it is found that the actual result is almost the same as the simulated one in the shape and size of ellipses and the

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Fig. 7. Computer simulation: (a) the simulated results with correction mass 1082.0gn2078 in plane A; (b) the simulated results with correction mass 802.0gn2178 in plane A. The thin lines represent the original vibrations, and the bold lines represent the simulated results.

azimuth of IPPs, confirming the balancing theory and the present method. 4.2. Multi-planes balancing The structural diagram of a 330 MW turbo-generator unit is shown in Fig. 1. As shown clearly in Fig. 9, the thin lines structure the 3D-holospectrum of original unbalance responses in bearing No. 1 to No. 4, in which higher vibrations focus on the bearing No. 1 and No. 3. Through analysis and diagnosis prior to field balancing, other rotating frequency faults could be excluded and it is safe to diagnose the leading fault as the mass unbalance, while force unbalance and coupling unbalance exist

Fig. 8. Balancing results. The thin lines represent the original vibrations, and the bold lines represent the residual vibration.

synchronously. Thus, the field balancing may be an effective method to reduce vibrations. The aim of field balancing is mostly focused on reducing the shaft vibrations of highpressure and intermediate-pressure cylinders. Three balancing planes were selected to mount the correction masses. Theses planes, marked in order as the plane A, B and C, locate respectively in the No. 1 bearing flange, the rigid coupling between bearing No. 2 and No. 3 and the rigid coupling between bearing No. 4 and No. 5. According to all previous data of trial runs, the transfer matrices of balancing planes with unit trial weight 1000gn08 can be calculated, namely RA, RB and RC. Based on the transfer matrices RA, RB and RC, the balancing calculation method is similar as that of above-mentioned single plane balancing. The correction mass is added on A plane to cancel the axial vibration on bearing section No. 1 as shown in Fig. 9a, and another correction mass is added on plane B to cancel the axial vibration on bearing section No. 3 as shown in Fig. 9b, while the correction mass added on C plane is used to cancel the axial vibration on bearing section No. 4 as shown in Fig. 9c. In Fig. 9a–c, the bold lines respectively represent the simulated vibration responses caused by pure correction masses on plane A, B and C, in which all the IPPs display a mirror symmetry relationship with those in original vibration 3Dholospectrum. By confirmation of the azimuths of correction masses in three balancing planes, GA was used to optimize the magnitude of correction masses, and the optimized results are QA = 548.6gn1168 in plane A, QB = 937.0gn608 in plane B, and QC = 1088.0gn628 in plane C. After on-site adding the optimized correction masses, the turbo-generator unit starts up anew, and the residual vibrations are measured. The 3Dholospectrum of the residual vibrations is shown by bold lines in Fig. 9d. It is clear that compared with the original vibrations, the all-residual axial vibrations after field balancing are far below the alarm value, and the vibration distribution along axial direction is more uniform, indicating successful balancing.

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Fig. 9. Multi-planes balancing of rotor system: (a) the simulated vibration responses due to pure correction mass QA = 548.6gn1168 in plane A, where IPP of the correction mass ellipse displays a mirror image of IPP of the original vibration ellipse in bearing section No. 1; (b) the simulated vibration responses due to pure correction mass QB = 937.0gn608 in plane B, where IPP of the correction mass ellipse displays a mirror image of IPP of the original vibration ellipse in bearing section No. 3; (c) the simulated vibration responses due to pure correction mass QC = 1088.0gn628 in plane C, where IPP of the correction mass ellipse displays a mirror image of IPP of the original vibration ellipse in bearing section No. 4. The thin lines represent the original vibrations, and the bold lines represent the vibration responses due to pure correction mass added in each balancing plane.

5. Conclusions 1. Based on the holospectral technique, this paper presents a new balancing method for rotor systems, named the holobalancing method, which fully utilizes the information from all sensors. Compared with traditional balancing methods, it reduces errors in balancing calculation due to limited utilization of the information from a unidirectional sensor, and thus improves balancing accuracy. 2. The presentation of the phase shift ellipse and the initial phase point is the key value of holo-balancing. It improves the theoretical basis, foresight, and safety in balancing process. 3. The transfer matrix is used to illuminate vibration responses in all bearing sections caused by unit trial weight, which can describe vibration characteristics of unbalance more accurately than traditional influence coefficient method.

4. The paper establishes correspondences between the angle of revolution and the angle of rotation. The angle compensation is presented to realize transformation between these two angles, which is a key technique for improving balancing accuracy. 5. Holobalancing technique synthetically utilizes techniques like information integration, computer simulation, and genetic algorithm optimization. It successfully combines classical techniques with the high and new ones. The synthetic advantage of holobalancing technique is that it increases foresight, safety in balancing procedure, but decreases the number of test runs and wastages. Acknowledgements The authors acknowledge supports from the National Education Ministry Doctor Foundation of China (No. 9269830, No. 20040698017), and the National Science

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Foundation of China (No. 50335030, No. 50475084). Special thanks are extended to Puchen Power Plant, Weihe Power Plant, and Shijiazhuang Oil Refinery, who provided the experimental data in the paper. References [1] L. Qu, H. Qiu, G. Xu, Rotor balancing based on holospectrum analysis principle and practice, China Mech. Eng. 19 (1998) 60–63. [2] L. Qu, X. Liu, G. Peyronne, Y. Chen, The holospectrum: a new method for rotor surveillance and diagnosis, J. Mech. Syst. Signal Process. 3 (1989) 255–267.

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[3] L. Qu, G. Xu, One decade of holospectral technique: review and prospect, in: Proceedings of the 1999 ASME Design Engineering Technical Conferences, 1999. [4] S. Liu, L. Qu, Vibration failures due to the non-uniform thermal deformation diagnosed by the use of hologram spectral techniques, J. Eng. Therm. Energy Power 19 (2004) 343–346. [5] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading MA, 1989. [6] B. Xu, L. Qu, X. Tao, The information integration and optimization in flexible rotor balancing, J. Sound Vib. 238 (2000) 877–892. [7] M. Srinvas, L.M. Patnaik, Adaptive probabilities of crossover and mutation genetic algorithms, IEEE Trans. SMC 24 (1994) 656–666.