Study on Vibration Response Reduction of Bladed Disk by Use of Asymmetric Vane Spacing

(Study on Response Reduction of Mistuned Bladed Disk)

Yasutomo Kaneko1, Masaki Ohta1, Kazushi Mori2 and Hiroharu Ohyama2

1 Department of Mechanical and Systems Engineering Faculty of Science and Technology, Ryukoku University 1-5 Yokotani, Ohe, Seta, Ohtsu, Siga 520-2194, JAPAN

2 Mitsubish Heavy Industries, Ltd.

ABSTRACT It is well known that asymmetric vane spacing can result in decreased levels of the excitation at specific frequencies. In this paper, the resonant response reduction of mistuned bladed disks due to asymmetric vane spacing is studied theoretically for the most probable asymmetric vane, in which the vane count of the upper and lower half is slightly different. First, a method for predicting the maximum amplitude of the mistuned bladed disk for the asymmetric vane spacing is proposed. Second, a parametric study is carried out using Monte Carlo simulation to clarify the vibration response characteristics of the mistuned bladed disk for the asymmetric vane spacing. From these results, it is concluded that asymmetric vane spacing is effective for reduction of resonant amplitudes of a mistuned bladed disk if a multi-resonance phe-nomenon does not appear. INTRDUCTION

In a multi-stage turbo-machinery, the interaction between the vane and the blade generates the excitation force on the blade, which comes from the wake of the upstream vane or the potential field of the upstream/downstream vane. The fundamental fre-quency of the excitation force due to the interaction between the vane and the blade is the rotor speed multiplied by the vane count, and if the natural frequency of the blade is coincident with the frequency of the excitation force, the resonant stress of the blade may become very large and may cause a blade failure due to HCF.

Besides the traditional methods to reduce the resonant stress, it is well known that the asymmetric vane spacing can result in de-creased levels of the excitation force at specific frequencies [1, 2]. It seems that asymmetric vane spacing is a very effective method for controlling the blade response for variable speed engines where the axial rotor span cannot be lengthened.

As for a tuned bladed disk, where dynamic properties of indi-vidual blades on a disk is the same, the reduction effect of the resonant stress due to asymmetric vane spacing has been studied theoretically and experimentally [3, 4]. However, realistic bladed disks, although designed mostly to be cyclically symmetric, are mistuned, i.e. dynamic properties of individual blades are different. In spite that such imperfections of mistuned bladed disks are usu-ally small, it is well known that the loss of cyclic symmetry prop-erties of the bladed disk by mistuning can drastically change the forced response levels. There are numerous papers studying the mistuning effect. In the early papers [5, 6], the forced response of mistuned bladed disk for symmetric vane spacing has been studied

by use of simple spring-mass model. Modern methods of the nu-merical analysis are described in the recent papers [7-9]. However, as far as authors know, few studies have been carried out on the reduction effect of the forced response of the mistuned bladed disk due to asymmetric vane spacing.

In this paper, the resonant response reduction of mistuned bladed disks due to asymmetric vane spacing is studied theoretically for the most probable asymmetric vane, in which the vane count of the upper and lower half is slightly different. First, a method for pre-dicting the maximum amplitude of the mistuned bladed disk for the asymmetric vane spacing is proposed. Second, a parametric study is carried out using Monte Carlo simulation to clarify the vibration response characteristics of the mistuned bladed disk for the asymmetric vane spacing. From these results, it is concluded that asymmetric vane spacing is effective for reduction of resonant amplitudes of a mistuned bladed disk. And except for the mul-ti-resonance phenomena, the maximum response of the mistuned bladed disk to asymmetric vane spacing can be roughly predicted by the maximum response of the mistuned bladed disk to symmet-ric vane spacing multiplied by the response reduction effect for free-standing blade without circumferential coupling [3]. ANALYSIS METHOD

As for the asymmetric vane spacing, usually the whole vanes in the stage are equally divided into Ns segments, and the following techniques are used to make asymmetric vane spacing. (1) Vane count in Ns segments is slightly altered (Vane count

method). (2) Vane count in Ns segments is equal but the vane pitch between

segments is changed by shifting segments circumferentially with respect to each other (Tangential offset method).

(3) All vane pitches are altered at random (Random pitch method). Although these methods are well known, it seems that only the

vane count method, where the whole vanes are divided into two segments (Ns = 2, upper half and lower half) and the vane count in the upper and lower half is slightly altered, is most practical and has been applied to actual engines, taking the cost of manufacturing and maintenance, disadvantageous effect on the excitation force of the lower engine orders, and the stability of the flow such as surge into consideration. Therefore, in this paper, the analytical method for predicting the blade vibration response of the asymmetric vane spacing by use of vane count method (Ns = 2) is mainly described, but the same procedure can be used for other methods.

Blade-Vane Interaction Force for Asymmetric Vane Spacing

When the vane counts in the upper and lower half are N1 and N2, respectively, two kinds of blade-vane interaction analyses with 3D

International Journal of Gas Turbine, Propulsion and Power Systems February 2012, Volume 4, Number 1

Copyright © 2012 Gas Turbine Society of Japan

Presented at International Gas Turbine Congress 2011 Osaka, November 13-18, Osaka, Japan, IGTC2011-0065 Review completed February 7, 2012

35

CFD are first carried out for the symmetric vane spacing, where the whole vane counts are 2N1 and 2N2. Then, the pressure fluctuation on the blade surface is obtained in the form of the time-history wave by connecting both results. Finally, the magnitude and phase of the pressure fluctuation on the blade surface corresponding to each harmonic component (engine order) are calculated by Fourier analysis of the time-history wave as expressed by Eq.(1). � � � � � � � � � � � � � � � � � � � � � � � � �

(1) Where, {F(t)} denotes the excitation force on the nodes of the blade surface caused by the blade-vane interaction, � the rotation speed of the rotor, H the harmonic (engine order) of the excitation force, �=H� the excitation frequency. In this study, the vibration re-sponse to each engine order is calculated separately by the modal analysis method described in the next section, and the steady re-sponse to the asymmetric vane spacing is obtained by summing up the calculated results. The direct current component of the excita-tion force in Eq.(1) is neglected in the vibration response analysis, because it has no influence on the vibration response. Vibration Response Analysis of Mistuned Bladed Disk by Modal Analysis

A bladed disk with continuous ring type structure, where the blades are continuously coupled at shrouds by blade untwist due to centrifugal force when the blades rotate at high speed, is modeled by the equivalent sprig-mass model in Fig. 1. Hereafter, the bladed disk with continuous ring type structure is referred to as ISB (In-tegral Shroud Blade). In Fig. 1, m1, k1 and c are the equivalent mass, stiffness and damping coefficient of the blade, respectively, while m2 and k3 are the equivalent mass and stiffness of the disk, respectively. k2 is the coupling stiffness of the shroud, and k4 is the coupling stiffness of the disk in the circumferential direction. The variable with superscript i means that the values for individual blades are slightly different. In the analysis of the FSB (free-standing blade) structure without shroud and the analysis of the GB (grouped blades) structure where a few blades are con-nected by shroud, the same equivalent model can be used, if the coupling stiffness k2 is set to be zero between blades or groups. When the bladed disk in Fig. 1 is rotating under the excitation force of Eq.(1), the equation of the motion of the bladed disk is expressed by Eq.(2). The damping term is dropped in Eq.(2), because the modal damping is introduced into the equation of the motion later.

(2)

Fig. 1 Equivalent spring mass model of bladed disk

Where,

(3) In Eq.(3), {xi} denotes the displacement vector of i-th blade, and {x0} = {xN} and {xN+1} = {x1} are satisfied, where N is the blade count of the whole bladed disk. {fe} is the vector of the modal excitation force on the blade, which can be calculated by the blade-vane interaction force {PH} in Eq.(1) and the vibration mode calculated by FEA (Finite Element Analysis). � denotes the angular frequency of the excitation force, and ��i is the phase angle caused by the rotation of the bladed disk, which can be expressed by Eq.(4).

(4)

Assuming the solution of Eq.(2) by Eq.(5), and applying the conventional modal analysis method to Eq.(2), the frequency re-sponse to the excitation engine order H is obtained as Eq.(6). (5) (6) Where, (7) In Eq.(6), {��r} and �r are the eigenvector and the natural fre-quency of the bladed disk obtained from the eigenvalue equation, in which the excitation force in Eq.(2) is set to be zero. kr denotes the modal stiffness, and��r the modal damping ratio, where the sub-script r is the order of the vibration modes.

When the mistuned bladed disk is rotating through symmetric vane spacing, the vibration response characteristics is described in Fig.2(a), where the dominant engine order of the excitation force is just one, H. Therefore, the vibration response of the bladed disk can be calculated directly by Eq.(6). On the other hand, when the mistuned bladed disk is rotating through asymmetric vane spacing, the vibration response characteristics is represented by Fig.2(b), where the bladed disk is excited by a few dominant engine orders. Therefore, first, the vibration response to each engine order is calculated by Eq.(6), and then, the vibration response to the asymmetric spacing is obtained by summing up the results. In brief, the mistuned bladed disk rotating through asymmetric vane spacing is obtained as follows. (1) The blade-vane interaction force is obtained by 3D CFD for a

bladed disk rotating through asymmetric vane spacing, and the amplitude of the excitation force corresponding to each engine order is calculated by Fourier analysis. In this study, for sim-plicity, the blade-vane interaction force is assumed by Eq.(8), and is expanded to Fourier series.

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(8) (2) Based on the information obtained from FEA results and ex-

perimental data, the modal parameters of the tuned bladed disk (m1, m2, k1, k2, etc.) are determined in such a manner that the natural frequency and the logarithmic decrement of the analysis model become equal to those of an actual bladed disk for the nodal diameter mode of interest..

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Mistuned bladed disk is assembled using k1i .

(4) Frequency response of a mistuned bladed disk to each engine order is calculated by Eq.(6), and then the frequency response of the bladed disk rotating through asymmetric vane spacing is obtained by summing up the results as shown in Fig. 2.

(5) After many repeated calculations of procedure (3) and (4) (one hundred times in this study), the calculated results are processed statistically. The maximum amplitude and the standard deviation of the vibration response of the mistuned bladed disk are evalu-ated (Monte Carlo simulation).

(a) Response to symmetric vane

(b) Response to asymmetric vane Fig. 2 Typical response of mistuned bladed disc to symmetric

and asymmetric vane ANALYSIS RESULTS

Frequency response analysis of a mistuned bladed disk rotating through symmetric and asymmetric vane spacing (vane count method, Ns=2) was carried out according to procedure described in the previous section. First, in order to compare the vibration re-

sponse characteristics of the FSB and ISB structure to asymmetric vane spacing, which are the typical bladed disks used in tur-bomachinery, the frequency response analysis was carried out for two kinds of models. The parameters of bladed disks analyzed are shown in Table 1. The modal parameters of bladed disks in Table 1 are determined from the information from FEA results and ex-perimental data. Figure 3 shows the natural frequency of the bladed disks shown in Table 1.

Table 1 Parameters of bladed disk analyzed

Fig. 3 Natural frequency of tuned bladed disk Analysis Results of bladed disk with FSB structure

Vibration response of FSB structure to symmetric vane spacing. In order to get the reference data for comparison, first, the vibration response analysis of bladed disks to symmetric vane spacing was carried out. One hundred mistuned bladed disks were assembled according to the procedure explained in the previous section, and the frequency responses of all bladed disks were cal-culated. Processing the calculated results statistically, the maxi-mum value and the distribution of the responses were evaluated.

Figure 4 shows the typical result of the frequency response analysis of the 20th engine order for the FSB structure, where the standard deviation of the blade alone frequency ( )/ bf f# is 3 %. In Fig. 4, frequency responses of all blades on a disk (80 blades) are superposed. Figure 5 shows the histogram of the maximum am-plitude for all blades on one hundred mistuned bladed disks (80 blades times 100 disks). The ordinates of Fig. 4 and the abscissa of Fig.5 are normalized by the maximum response (the resonant am-plitude) of the tuned bladed disk.

Figure 6 shows the relationship between the standard deviation of the blade alone frequency and the maximum and mean amplitude of the mistuned bladed disks. In Fig.6, the coefficient of the varia-tion of the response (the ratio of the standard deviation and the mean value) is also plotted. As shown in Fig. 4 to Fig. 6, the FSB structure is very sensitive to the mistuning. If the frequencies of the blades on a disk are slightly different, the responses of the blades change drastically, and the maximum amplitude reaches nearly twice that of the tuned bladed

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disk as shown in Fig. 5 and Fig. 6. The maximum amplitude reaches its maximum value in the vicinity of 0.5 % standard deviation of the blade alone frequency, and if the standard deviation of the blade alone frequency becomes greater than this, the maximum amplitude begin decreasing as shown in Fig. 6. On the other hand, the mean amplitude of the mistuned bladed disk reaches its maximum at 0 % standard deviation of blade alone frequency (at the tuned bladed disk). In other words, the mean amplitude of the mistuned bladed disk is always smaller than that of the tuned bladed disk. This is because in the mistuned bladed disk, large amplitudes appear only in a few specific blades, while in the most other blades, the re-sponse becomes smaller than that of the tuned bladed disk.

Fig. 4 Frequency response of mistuned bladed disk (�f = 3 %, H=20)

Fig.5 Histogram of blade amplitude (H=20)

Fig. 6 Frequency deviation and maximum amplitude (H=20)

The results of vibration response analysis to the 72nd engine order are shown in Fig. 7 to Fig. 9. Figure 7 shows the typical result of the frequency response analysis of the 72nd engine order for the FSB structure, where the standard deviation of the blade alone frequency ( )/ bf f# is 3 %. In Fig. 7, frequency responses of all blades on a disk (80 blades) are superposed. Figure 8 shows the histogram of the maximum amplitude for all blades on one hundred mistuned bladed disks (80 blades times 100 disks).

Figure 9 shows the relationship between the standard deviation of the blade alone frequency and the maximum and mean amplitude

of the mistuned bladed disks. In Fig.9, the coefficient of the varia-tion of the response (the ratio of the standard deviation and the mean value) is also plotted. Roughly evaluating, the vibration response characteristics of the mistuned bladed disk to the excita-tion force of the 72nd engine order is the nearly same as that to the excitation force of the 20th engine order. In the frequency response to the 72nd engine order, however, the maximum amplitude reaches its maximum value in the vicinity of 1.0 % standard deviation of blade alone frequency, and the maximum amplitude reaches more than twice that of the tuned bladed disk as shown in Fig. 9. The difference of the vibration response characteristics of the mistuned bladed disk due to the engine order of the excitation force is closely related to the characteristics of the natural frequency of the bladed disk shown in Fig. 3 [5, 6]. The feature of the calculated frequency response of the mistuned bladed disk for symmetric vane spacing is consistent with those of published papers.

Fig.7 Frequency response of mistuned bladed disk (�f = 3 %, H=72)

Fig. 8 Histogram of blade amplitude (H=72)

Fig.9 Frequency deviation and maximum amplitude (H=72)

Vibration response of the FSB structure to asymmetric vane spacing. Figure 10 shows the result of the frequency response analysis to the asymmetric vane spacing (vane count of the upper half N1=11�vane count of the lower half N2=9) for the FSB structure, where the standard deviation of the blade alone frequency ( )/ bf f# is 3 %. In Fig. 10, frequency responses of all blades on a

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JGPP Vol. 4, No. 1

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disk (80 blades) are superposed. In Fig. 10, the abscissa is the rotation speed, and the ordinate is normalized by the maximum response (the resonant amplitude) of the tuned bladed disk to the excitation force of the 20th (H=N1 +N2) engine order.

Figure 11 shows the histogram of the maximum amplitude for all blades on one hundred mistuned bladed disks (80 blades times 100 disks). The abscissa of Fig. 11 is also normalized by the maximum response (the resonant amplitude) to the excitation force of the 20th engine order of the tuned bladed disk. Figure 12 shows the rela-tionship between the standard deviation of the blade alone fre-quency and the maximum and mean amplitude of the mistuned bladed disks. In Fig.12, the coefficient of the variation of the re-sponse (the ratio of the standard deviation and the mean value) is also plotted. Comparing Fig. 4 with Fig. 10, it can be said that because the bladed disk rotating through asymmetric vane spacing is excited by excitation forces caused by a few engine orders, the number of the peaks in the frequency responses is larger than that to symmetric vane spacing, and the response characteristics become very com-plicated. As shown in Fig. 11 and Fig. 12, however, the maximum and mean amplitude of the mistuned bladed disk is decreased by

Fig. 10 Frequency response of mistuned bladed disk (�f = 3 %, N1=11, N2=9)

Fig. 11� Histogram of blade amplitude (N1=11, N2=9)

Fig. 12 Frequency deviation and maximum amplitude (N1=11, N2=9)

40 % to 50 %, comparing to those to symmetric vane spacing. That is, comparing Fig. 6 with Fig. 12, it is said that although the general tendency of the maximum amplitude, the mean amplitude and the coefficient variation to the frequency deviation is the nearly same for both of symmetric and asymmetric vane spacing, the absolute value of responses to asymmetric vane spacing is decreased by 40 % to 50 %. This is because although the bladed disk through asymmetric vane spacing is excited by a few engine orders, each excitation force is smaller than that caused by symmetric vane spacing. Therefore, even if the response to each engine order is summed up as shown in Fig. 2(b), the total response to asymmetric vane spacing is less than that to symmetric vane spacing, unless the multi-resonances occur.

In reference [3], the response reduction effect by asymmetric vane spacing for the FSB structure without the circumferential coupling) is summarized as shown in Fig. 13. In Fig. 13, the re-sponse reduction effect is calculated for a free-standing blade, where the flexibility of the disk (the coupling of a bladed disk in the circumferential direction) is neglected. The abscissa of Fig. 13, �Nvane denotes the product of the whole vane count (Nvane=N1+N2) and the logarithmic decrement of the blade �. Because the value of �Nvane corresponding to Fig. 12 is 0.2 (Nvane=20, logarithmic dec-rement �=0.01), the response reduction effect for the condition of Fig. 12 is estimated to be 0.48 from Fig.13. Consequently, for the FSB structure where the circumferential coupling is weak and the multi-resonances hardly occur, it is concluded that the maximum response of the mistuned bladed disk through asymmetric vane spacing (Fig.12) can be estimated by the maximum response of the mistuned bladed disk to symmetric vane spacing (Fig. 6) multiplied by the response reduction effect in Fig. 13 (1-R=0.52).

Fig. 13� Reduction effect of asymmetric vane [3]

Figure 14 shows the result of the frequency response analysis to the asymmetric vane spacing (vane count of upper half N1=37�vane count of lower half N2 =35) for the FSB structure, where the standard deviation of the blade alone frequency ( )/ bf f# is 3 %. In Fig. 14, frequency responses of all blades on a disk (80 blades) are superposed. In Fig. 14, the abscissa is the rotation speed, and the ordinate is normalized by the maximum response (the resonant amplitude) of the tuned bladed disk to the excitation force of the 72nd (H=N1 +N2) engine order. Figure 15 shows the histogram of the maximum amplitude for all blades on one hundred mistuned bladed disks (80 blades times 100 disks). The abscissa of Fig. 15 is also normalized by the maximum response (the resonant amplitude) to the excitation force of the 72nd engine order of the tuned bladed disk. Figure 16 shows the relationship between the standard devia-tion of the blade alone frequency and the maximum and mean amplitude of the mistuned bladed disks. In Fig.16, the coefficient of the variation of the response (the ratio of the standard deviation and the mean value) is also plotted.

Roughly evaluating, the vibration response characteristics of

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JGPP Vol. 4, No. 1

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the mistuned bladed disk to asymmetric vane spacing of Nvane =72 (Fig. 16) is the nearly same as that of Nvane =20 (Fig. 12). As for asymmetric vane spacing of Nvane =72, the value of �Nvane in Fig. 13 is 0.72, and therefore, the response reduction effect is evaluated to be 0.40 from Fig.13. Consequently, for the case of Nvane =72, it is concluded that the maximum response of the mistuned bladed disk through asymmetric vane spacing (Fig.16) can be estimated by the maximum response of the mistuned bladed disk to symmetric vane spacing (Fig. 9) multiplied by the response reduction effect in Fig. 13 (1-R=0.60).

Fig. 14 Frequency response of mistuned bladed disk (�f = 3 %, N1=37, N2=35)

Fig. 15 Histogram of blade amplitude (N1=37, N2=35)

Fig. 16 Frequency deviation and maximum amplitude (N1=37, N2=35)

Analysis Results of bladed disk with ISB structure. Response

analysis of the ISB structure was carried out in the same procedure as that of the FSB structure. First, the vibration response analysis of bladed disks to symmetric vane spacing was carried out to get the reference data. One hundred mistuned bladed disks were assembled, and the frequency responses of all bladed disks were calculated. Processing the calculated results statistically, the maximum value and the distribution of the responses were evaluated.

Figure 17 shows the result of the frequency response analysis of the 72nd engine order for the ISB structure, where the standard

deviation of the blade alone frequency ( )/ bf f# is 3 %. In Fig. 17, frequency responses of all blades on a disk (80 blades) are super-posed. Figure 18 shows the histogram of the maximum amplitude for all blades on one hundred mistuned bladed disks (80 blades times 100 disks). The ordinates of Fig. 17 and the abscissa of Fig. 18 are normalized by the maximum response (the resonant ampli-tude) of the tuned bladed disk.

Figure 19 shows the relationship between the standard devia-tion of the blade alone frequency and the maximum and mean amplitude of the mistuned bladed disks. In Fig. 19, the coefficient of the variation of the response (the ratio of the standard deviation and the mean value) is also plotted.

As shown in Fig. 19, the ISB structure is very insensitive to the mistuning. If the frequencies of the blades on a disk are different, the responses of the blades hardly change. The maximum response of the mistuned bladed disk becomes 20 % larger than that of the tuned bladed disk at most in the practical range of the standard deviation of the blade alone frequency. The feature of the calculated frequency response of the mistuned bladed disk with the ISB structure for symmetric vane spacing is also consistent with the fact pointed by many published papers.

Fig. 17 Frequency response of mistuned bladed disk (ISB, �f = 3 %, H=72)

Fig. 18 Histogram of blade amplitude (ISB, H=72)

Fig. 19 Frequency deviation and maximum amplitude

(ISB. H=72)

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JGPP Vol. 4, No. 1

40

Figure 20 shows the result of the frequency response analysis to the asymmetric vane spacing (vane count of upper half N1=37�vane count of lower half N2 =35) for the ISB structure, where the stand-ard deviation of the blade alone frequency ( )/ bf f# is 3 %. In Fig. 20, frequency responses of all blades on a disk (80 blades) are superposed. In Fig. 20, the abscissa is the rotation speed, and the ordinate is normalized by the maximum response (the resonant amplitude) of the tuned bladed disk to the excitation force of the 72nd (H=N1 +N2) engine order. Figure 21 shows the histogram of the maximum amplitude for all blades on one hundred mistuned bladed disks (80 blades times 100 disks). The abscissa of Fig. 21 is also normalized by the maximum response (the resonant amplitude) to the excitation force of the 72nd engine order of the tuned bladed disk. Figure 22 shows the relationship between the standard devia-tion of the blade alone frequency and the maximum and mean amplitude of the mistuned bladed disks. In Fig. 22, the coefficient of the variation of the response (the ratio of the standard deviation and the mean value) is also plotted.

Fig. 20 Frequency response of mistuned bladed disk (ISB, �f = 3 %, N1=37, N2=35)

Fig. 21 Histogram of blade amplitude (ISB, N1=37, N2=35)

Fig. 22 Frequency deviation and maximum amplitude

(ISB. N1=37, N2=35)

For the ISB structure with large blade damping (�=0.03), the value of �Nvane in Fig. 13 is 2.2 for asymmetric spacing of Nvane=72, and therefore, the response reduction effect is evaluated to be 0.22 from Fig. 13. Consequently, for the ISB structure, it is also con-cluded that the maximum response of the mistuned bladed disk through asymmetric vane spacing (Fig.22) can be estimated by the maximum response of the mistuned bladed disk to symmetric vane spacing (Fig. 19) multiplied by the response reduction effect in Fig. 13 (1-R=0.78).

On the other hand, Fig. 23 shows the relationship between the standard deviation of the blade alone frequency and the maximum and mean amplitude of the mistuned bladed disks for symmetric vane spacing (H =20) and asymmetric vane spacing (vane count of the upper half N1=11�the lower half N2=9), respectively. As shown in Fig. 23, if the frequencies of the blades on a disk change, the responses of the blades hardly change. Moreover, the maximum response to asymmetric vane spacing becomes slightly larger than that to symmetric vane spacing if the frequency deviation is beyond 2 %. This is because for the ISB structure analyzed, the multi- resonances happen to appear around the resonance point to the 20th engine order as shown in Campbell diagram of Fig. 24, where the resonance points satisfying resonance condition [10] are plotted. Therefore, if the responses to each engine order are summed up, the total amplitude becomes larger than that to symmetric vane spacing. Consequently, it is pointed out that when asymmetric vane spacing is applied for the ISB structure, it is essential to check whether there are multi-resonances within operation range. Fig. 23 Frequency deviation and maximum amplitude of ISB for � � Symmetric (H=20) and asymmetric (N1=11, N2=9) vane

Fig.24 Campbell diagram of 1st mode family of ISB

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JGPP Vol. 4, No. 1

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Analysis Results of bladed disk with GB structure Figure 25, Fig. 26 and Fig. 27 show the analysis results of the GB

structure. The number of the blades of the GB structure analyzed is 120 (N =120), and the bladed disk consists of the 10 groups of 12 blades. The analysis model of this GB structure is the same as that in the reference [11]. Figure 25 shows the natural frequency, and Fig. 26 shows the Campbell diagram of the bladed disk, in which the resonance points satisfying the resonant condition [10] are plotted. Figure 27 shows the relationship between the standard deviation of the blade alone frequency and the maximum amplitude of the mistuned bladed disks to the symmetric and asymmetric vane spacing.

As shown in Fig. 27, the response reduction effect of the GB structure analyzed is small even for the tuned bladed disk, because the multi-resonances happen to appear around the resonance point to the 26th engine order as shown in Fig. 26. In addition, when the standard deviation of the blade alone frequency is larger than 5 %, the maximum response of the mistuned bladed disk to the asym-metric vane spacing becomes lager than that to the symmetric vane spacing. This is also caused by the multi-resonances around the resonance point to the 26th engine order.

Fig. 25 Natural frequency of tuned grouped bladed disk

Fig. 26 Campbell diagram of tuned grouped bladed disk

Fig. 27 Maximum response of bladed disk with GB structure

CONCLUSION In this study, using the equivalent spring mass model of the

bladed disk, the resonant response analysis of the mistuned bladed disk was carried out, and the resonant response reduction due to asymmetric vane spacing was studied with Monte Carlo method. From these results, it is concluded that for the mistuned bladed disk, the asymmetric vane spacing is effective to reduce the resonant response, and the maximum response of the mistuned bladed disk rotating through asymmetric vane spacing can be roughly estimated by the maximum response of the mistuned bladed disk to symmet-ric vane spacing multiplied by the response reduction effect for free-standing blade without the circumferential coupling, unless the multi-resonances happen to appear. On the other hand, if the mul-ti-resonances occur, it is possible that the asymmetric vane spacing increases the resonant response level. Therefore, it is pointed out that when asymmetric vane spacing is applied for the ISB or GB structure, in which the multi-resonances could appear due to strong coupling of the bladed disk in the circumferential direction, it is essential to check whether there are multi-resonances within oper-ation range. REFERENCES [1] Kemp, R. H., Hirschberg, M. H. and Morgan, W. C., “Theoret-

ical and experimental analysis of the reduction of rotor blade vibration in turbomachinery through the use of modified stator vane spacing”, NACA technical note, 4373 (1958), pp. 1-43.

[2] Clark, J. P., Aggarwala, A. S., Velonis, M. A., Gacek, R. E., Magge, S. S. and Price, F. R., “Using CFD to reduce resonant stresses on a single-stage, high-pressure turbine blade”, ASME GT-2002-30320, (2002).

[3] Kaneko, Y. et al., “Reduction of vibratory stress of compressor blade by use of asymmetric vane spacing (in Japanese)”, Transactions of the Japan Society of Mechanical Engineering, Series C, Vol.71, No.712 (2005), pp. 3409-3416.

[4] Kaneko, Y. et al., “Reduction of resonant stress of turbine blade by use of asymmetric vane spacing (in Japanese)”, Transactions of the Japan Society of Mechanical Engineering, Series C, Vol.72, No.720 (2006), pp. 2366-2372.

[5] Kaneko, Y. et al., “Vibration response analysis of mistuned bladed disk (in Japanese)”, Transactions of the Japan Society of Mechanical Engineering, Series C, Vol.58, No.547 (1992), pp. 744-749.

[6] S. T. Wei and C. Pierre, “Localization phenomena in mistuned assemblies with cyclic symmetry part 1: Free vibrations”, Transactions of ASME, J. of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 110, (1988), pp. 429-438.

[7] Petrov, E.P., Sanliturk, K. Y. and Ewins, D. J., “A new method for dynamic analysis of mistuned bladed discs based on exact relationship between tuned and mistuned systems”, Transac-tions of ASME, J. of Engineering for Gas Turbines and Power, Vol.122, (2002), pp. 886-597.

[8] Bladh, R. and Pierre, C., “Dynamic response prediction for a mistuned industrial turbomachinery rotor using reduced-order modeling”, Transactions of ASME, J. of Engineering for Gas Turbines and Power, Vol.124, (2002), pp. 942-952.

[9] Feiner, D. M. and Griffin J. H., “A fundamental model of mistuning for a single family of modes”, Transactions of ASME, J. of Engineering for Gas Turbines and Power, Vol.124, (2002), pp. 597-605.

[10] Namura, K., “The vibration characteristics of steam turbine blades : 2nd report (in Japanese)”, Transactions of the Japan Society of Mechanical Engineering, Series C, Vol.52, No.474 (1986), pp. 3409-3416.

[11] Kaneko, Y., “Vibration characteristics and mistuning effect of grouped blades (in Japanese), Proceedings of JSME, No.954-5 (1995), pp. 182-184.

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