American Institute of Aeronautics and Astronautics

1

Fatigue Strength of Rocket Pump Inducers

Masaharu UCHIUMI* and Kenjiro KAMIJO.† Japan Aerospace Exploration Agency, Kakuda, Miyagi, 981-1526, Japan

Norio SAKAZUME‡ Japan Aerospace Exploration Agency, Tanegashima, Kagoshima, 891-3793, Japan

and

Rei MIHARA§ Ishikawajima-Harima Heavy Industries Co., Ltd., Mizuho, Tokyo, 190-1297, Japan

Various kinds of unsteady phenomena, such as cavitation surge, rotating cavitation, rotating stall, backflow cavitation, etc., have occurred in recent rocket pump inducers for booster engines because they are used under the condition of high head, high load and low suction pressure. Therefore, it is very important to estimate unsteady stress on the blades to obtain the required durability of an inducer. An analytical method to obtain fatigue life, which utilizes the results of water tests, is presented in this paper.

Nomenclature D = cumulative fatigue damage E = Young’s modulus N = rotational speed Q = volume flow rate Q/Qd = flow ratio R = stress ratio α = conversion exponent of rotational speeds ⊿ε = variable strain range θ = circumferential angle κ = cavitation number ν = Poisson’s ratio ρ = density ⊿σ = variable stress range φ = flow coefficient of inducer inlet ω = frequency of shaft rotational speed Subscripts d = design point H2O = water test LOX = liquid oxygen test OTP = nominal operation of turbopump

* Associate Senior Engineer, Rocket Engine Technology Center and H-IIA Project Team, Member AIAA. † Invited Staff (Chief Engineer), Rocket Engine Technology Center, Senior Member AIAA. ‡ Associate Principal Engineer, Kagoshima Space Center. § Staff Engineer, Space Technology Group, Research & Engineering Division.

42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit9 - 12 July 2006, Sacramento, California

AIAA 2006-5072

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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I. Introduction T is very important to estimate unsteady stress on the blades to obtain the required durability of an inducer, because various kinds of unsteady phenomena, such as cavitation surge, rotating cavitation, rotating stall,

backflow cavitation, etc., have occurred in recent rocket pump inducers for booster engines.1-12 The fatigue life of an inducer was estimated using unsteady stress of experimental data of water flow tests and

static stress of blades obtained by the FEM analysis. The experiments were carried out using a helical inducer with three blades which was made for water tests. Unsteady strain generated by the cavitation was obtained by using strain gauges attached to the blade surfaces of the inducer. Variable stress on the root of the blades corresponding to estimated typical flight conditions was calculated by correcting rotational speeds of water tests using a conversion exponent of the rotational speed α, which was greatly influenced by cavitation instabilities, such as rotating cavitation, cavitation surge, etc..

A frequency distribution analysis of variable stress was conducted by the rain flow method with variable strain data of water tests, and the fatigue life was evaluated by the cumulative fatigue damage. The present study shows that the conversion exponent of the rotational speed α changes from 1.0 to 2.0 and strongly depends upon the cavitation instability. Operation duration with a large value of conversion exponent strongly affects the fatigue life of inducers. Therefore, it is very important to know the inlet flow condition of an inducer which changes momentarily during an actual flight.

II. Static Strength and Resonance of Inducer Blades

Main design parameters and typical condition of the inducer The inducer used in the present study is designed for liquid oxygen turbopump. It has three cambered swept-

back helical blades shown in Fig. 1. The design flow coefficient, the inlet incident angle and the tip solidity are 0.0775, 3.3° and 1.91, respectively. Blades near the hub are fairly thick for the purpose of increasing of the strength. This inducer is made from nickel based superalloy, Inconel 718. Table 1 shows the standard operating condition of the inducer during the typical flight.

Table 1 Standard operating condition of the inducer

Rotational speed, N 18,330 rpm Flow rate, Q 220.0 kg/s Inlet static pressure 0.735 MPa Inlet temperature 90.5 K

Table 2 Static stress of the blade by FEM analysis

Analysis of static strength The static strength of the inducer was estimated by the FEM analysis in consideration of the centrifugal force

and the static pressure distribution of blades which were obtained by CFD analysis. Validity of this static strength was confirmed by comparison between the static strain measured in water tests and the estimated stress by the FEM analysis. The structural model for this FEM is a three dimensional solid model. The material characteristics of Inconel 718 at 90 K are estimated by linear interpolation of cryogenic data. For this CFD analysis, a commercial code “STAR-CD” with incompressible NS equation was used out of consideration of the cavitation occurrence. The number of grids of this calculation is 910,000, which takes into consideration of tip clearance between inducer blades and a casing. The calculation was performed under the condition of inlet pressure of 0.735 MPa and inlet temperature of 90.5 K.

I

Fig. 1 Configuration of the inducer

Flow ratio, Q/Qd Static stress Circumferential angle(MAX), θ Margin of safety(*)

0.90 1168 MPa 88 deg 0.35 1.00 961 MPa 88 deg 0.65 1.10 770 MPa 88 deg 1.06

(*)Allowable primary membrane stress + primary bending stress

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The sum of primary membrane stress and primary bending stress obtained by the static strength analysis is shown in Table 2. The sum of these values are below 1,580 MPa, which is the design criteria of this inducer determined by considering the safety factor of Inconel 718. The margin of safety（ in Table 2 ）is the number that the ratio of the allowable stress to the estimated stress minus 1.0. Plus sign of the margin of safety means that this inducer has enough static strength. The stress is fairly high at the position of circumferential angle of 65~110 degrees and the maximum stress occurs at the position of 88 degrees as shown in Table 2, which is regardless of flow ratios. The stress increases with decrease of flow ratio, which is explained as follows: the decrease of the flow rate increases inducer head, which makes the load of blades larger.

The static stress obtained in water tests at the flow ratio of 0.94 ~ 1.06 were 630 ~ 800 MPa at the position of circumferential angle of 90 ~ 105 degrees. These values are slightly higher than calculated ones. Calculated and measured stress of the nominal flow rate are 961 MPa (at θ= 88) and 713 MPa (at θ= 90), respectively. The calculated stress is overestimated by 25 percent compared with measured stress.

Analysis of resonance of inducer blades

To avoid the resonance of inducer blades, the possibility of inducer blades resonance was investigated using Campbell diagram. The resonance frequencies of inducer blades were obtained by correction of centrifugal force, temperature and added mass to the frequencies measured by the holography method under the condition of room temperature and atmospheric pressure. Regarding the correction of centrifugal force and temperature, the correction factor was obtained by the result of the FEM analysis and the change of Young’s modulus, respectively. And the validity of the correction factor of added mass was verified by the measured natural frequencies in water.

The Campbell diagram is shown in Fig. 2. The axis of abscissas indicates rotational speed. The rotational speed of operation in the flight varies from 17,680 rpm to 19,360 rpm, which is indicated in Fig. 2. The axis of ordinate shows frequency. Three natural frequencies of the first, second and third, were obtained using the correction factors of temperature, centrifugal force and added mass. The frequencies of 1~3ω are far away from the three natural frequencies. That is, the resonance of low degrees of these inducer blades can be avoided during the flight operation. It is also clarified that the interference of the guide vane of the inducer outlet (7ω) and the three natural frequencies is avoided.

III. Analysis of Fatigue Life A flow diagram to estimate the fatigue life of the present inducer is shown in Fig. 3. The details of this

estimation are as follows:

Estimation of unsteady stress in actual operation by water tests The unsteady component of stress in the flight operation was obtained by correction of the measured variable

stress in water tests just the same as Rosenmann’s test,2 because it is difficult to estimate accurately by CFD analysis. Tests to measure the strain of the inducer blade surface were performed with lower rotational speed than that of the flight operation due to the limitation of the test facility. The conversion of the variable strain measured in water tests, ⊿εH2O, to the variable stress of the flight operation, ⊿σ, is performed by the following equation.

222

2 1∆∆

νE

ωω

ρρεσ

α

OH

OTP

OH

LOXOH −

×⎟⎟⎠

⎞⎜⎜⎝

⎛××= (1)

0

1000

2000

3000

0 5000 10000 15000 20000 25000

Rotating Speed (rpm)Fre

quen

cy (

Hz)

７ω

Min

. Spe

ed

Max

. Spe

ed

６ω

５ω

４ω

３ω

２ω

ω

third natural vibration

first natural vibration

second natural vibration

Fig. 2 Campbell diagram

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Fig. 3 Fatigue life analysis

where suffixes of H2O, LOX and OTP indicate water, liquid oxygen and the flight condition of the oxygen turbopump, respectively. An exponent of α in equation (1) is the correction factor of a rotational speed which was obtained as follows; the variable strain was measured in some tests with different rotational speeds ( 5,000 rpm ～9,000 rpm) which were performed under the same inlet flow condition of the inducer regarding the flow ratio Q/Qd and the cavitation number k, then, the frequency analysis of variable stress regarding test data of the different rotational speeds and the same cavitation number was conducted, lastly, the frequency distribution of the variable strain is obtained and compared by analyzing data of the variable strain with the rain flow method.

Because the rotating cavitation do not occur of this inducer with the casing which has a step near the leading edge to suppress the rotating cavitation,4,5 the effect of rotating cavitation on variable strain was investigated using another inducer which has a smaller incident angle than the present inducer. Calculated α were obtained in the rotating cavitation free and the rotating cavitation conditions, which are shown in Fig. 4 and Fig. 5, respectively. In the case of the rotating cavitation free condition, the curves of frequency distribution of 5,000 and 7,500 rpm are almost coincident, if the conversion exponent of rotational speeds α is assumed to be 1.0. Regarding Fig. 5, the conversion exponent is assumed to be 1.6. Therefore, it is concluded that α in the case of rotating cavitation condition is larger than that of the rotating cavitation free condition.

Figure 6 shows the result of FFT analysis under the rotating cavitation condition. The scale of the axis of abscissas regarding the data of 5,000 rpm is multiplied 1.5 to correspond to that of 7,500 rpm. The square of the measured variable stress with 5,000 rpm is shown in Fig. 6. Since the rotating speed frequency of this rotating cavitation is 150 Hz, the frequency of 25 Hz, which is recognized in stress fluctuations, comes from the difference between the rotating speed of inducer and that of the rotating cavitation. The component of the variable stress due to the rotating cavitation almost agrees with the square of a rotational speed ratio, which means α is 2.0. With significant instability phenomena like a rotating cavitation, it is considered that the conversion exponent increase to 2.0 since pressure difference between pressure and suction surfaces of a blade directly influence the variable stress.

Estimation of Variable Stress by Water Flow tests

Frequency Analysis of Variable Stress

Decision of the Profiles by the Flight Inlet Condition

Calculation of Fatigue Damage by the modified Miner’s

Evaluation of the Fatigue Life

Frequency [cycle]

0.0

0.2

0.4

0.6

0.8

1.0

1.00E+00 1.00E+02 1.00E+04 1.00E+06 1.00E+08

Var

iabl

e S

tress

Ran

ge Δ

σ／

Δσ

max

5,000rpm

7,500rpm

Fig. 4 Variable stress range by rotating speed conversion （α＝1.0）

Fig. 5 Variable stress range by rotating speed conversion（α＝1.6）

0.0

0.2

0.4

0.6

0.8

1.0

1.00E+00 1.00E+02 1.00E+04 1.00E+06 1.00E+08

Frequency [cycle]

Var

iabl

e S

tress

Ran

ge Δ

σ／

Δσ

max

5,000rpm

7,500rpm

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When the variable stress with wide-band is outstanding in the rotating cavitation free condition, the conversion exponent shows 1.0. It is considered that the conversion exponent of 1.6 in Fig. 5 is clearly due to an occurrence of the rotating cavitation. These means that conversion exponent changes between 1.0 and 2.0 according to the kind of cavitation instability.

Using the same method mentioned above, the conversion exponent of rotational speeds was investigated with the other inducer in which the cavitation surge occurred. The frequency of this cavitation surge was around 0.25ω. A result of the conversion exponent is shown in Fig. 7 in which variable stress ranges of rotational speeds of 5,000 rpm and 9,000 rpm were compared. Two curves of frequency distribution of the variable stress fairly well agree with each other if the conversion exponent is assumed to be 1.3.

Variable stress frequency distribution analysis

The frequency distribution analysis with the rain flow method and the correction of rotational speeds mentioned above was applied to the present inducer for inlet flow conditions during the actual flight (mentioned below).

Cumulative fatigue damage by modified Miner’s rule

The cumulative fatigue damage was calculated with modified Miner's rule to estimate the fatigue life of the present inducer. The fatigue damage is obtained by the sum of the ratio of the occurrence cycle ni, to the fatigue life to variable stress Ni,, which is expressed by the following equation.

∑= ii NnD (2) The data of fatigue strength of Inconel

718 forging at the cryogenic temperatures were obtained on the condition of constant stress ratio (R=0). And the fatigue strength diagram which was used in this calculation was estimated in consideration of the scatter of the data by three times the standard deviation, and the correction of the mean stress. It is shown in Fig. 8.

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800 1000

Frequency [Hz]V

aria

ble S

tress

Ran

ge Δ

σ／

Δσ

max

at 7,500 rpm

Square Conversion of Rotating Speed at 5,000 rpm

Fig. 6 FFT analysis of conversion factor of rotating speed

Fig. 7 Variable stress range by rotating speed conversion （α＝1.3）

100

1000

10000

1.0E+04 1.0E+06 1.0E+08

Cycles to Failure

Var

iable

Str

ess

Ran

ge　

Δσ

　Ｍ

Ｐａ

Measured Values by NIMS (R=0)

Mean Line（Approximation）

3-σ Line

Fig. 8 Inconel 718 forging fatigue strength diagram at 90K (LOX), R=0

0.0

0.2

0.4

0.6

0.8

1.0

1.00E+00 1.00E+02 1.00E+04 1.00E+06 1.00E+08

Frequency [cycle]

Var

iabl

e S

tress

Ran

ge Δ

σ／

Δσ

max 5,000rpm

9,000rpm

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In this procedure, to calculate the mean stress precisely, the static stress due to the centrifugal force was estimated by the FEM analysis, and the static stress due to the pressure difference between blade surfaces was estimated by utilizing the measured static strain obtained in water tests.

Estimation of Inlet flow condition of inducer during an actual flight

As mentioned in the previous section, the fatigue damage of inducer D is calculated using equation (2). To estimate the fatigue life of an inducer, it is necessary to know the inlet flow condition during an actual flight as follows; the rotational speed N, the flow ratio Q/Qd, the cavitation number κ, because the operation condition of an inducer momentarily changes correspondingly to time after the launch. In this study, these conditions were estimated by data in actual flights as shown in Fig. 9 for example.

Estimation of fatigue life

The cumulative fatigue damage of the present inducer was calculated considering three parameters in Fig. 9 for the operation duration of the inducer which includes two acceptance tests and four times of flight. It is considered that the fatigue failure of an inducer occurs when the calculated cumulative fatigue damage comes up to 1.Therefore, we will be able to know the margin of the fatigue life by the calculated fatigue damage. Since the distribution of blade load moves to the trailing edge with the decrease of a cavitation number, the maximum cumulative fatigue damage occurs at the position of the larger circumferential angle θ. Table 3 shows the cumulative fatigue damage of the present inducer.

According to Table 3, the operations of the maximum rotational speed and minimum flow ratio seem to cause larger and smaller fatigue damages, respectively. This can be explained by the longer operation duration with a large conversion exponent of rotational speed α due to the great change of cavitation numbers during estimated flight conditions shown in Fig. 9. However, it was clarified that the present inducer has enough fatigue life because all the calculated cumulative fatigue damage was below 0.25, which was required in design.

IV. Concluding Remarks Taking into consideration of unsteady cavitation phenomena, fatigue strength of a cavitating inducer was

evaluated. The fatigue life of the inducer was estimated by the measured variable stress of water tests. The following were obtained in this study.

1) Conversion exponent of a rotational speed α was obtained by using variable strain measured in water tests, which clarified that α shows 1.0 and 2.0 in the cases of the rotating cavitation free and the rotating cavitation condition, respectively.

2) The conversion exponent of a rotational speed is between 1.0 and 2.0, which depends upon the cavitation instability of an inducer.

3) We propose a new method to estimate the fatigue life using the cumulative fatigue damage derived from taking into consideration of three inlet flow conditions of an inducer as follows; the rotational speed, the flow ratio and the cavitation number. It is clarified that the operation duration with the large conversion factor of rotational speed α strongly influences the fatigue life of an inducer.

Table 3 Cumulative fatigue damage

Condition of flight history Cumulative fatigue damage

Nominal operation(Most probable) 0.039（θ＝75 deg） Maximum rotational speed 0.044（θ＝75 deg） Minimum flow ratio 0.035（θ＝120 deg）

0.0

0.3

0.6

0.9

1.2

1.5

0 100 200 300 400

Flight Time　[sec]

Q/Q

d、ｋ*10

　[－

]

15000

16000

17000

18000

19000

20000

Rota

ting

Speed

N　

[rpm

]

Q/Qd

N

ｋ

Fig. 9 An Example of Estimated Inlet flow Conditions of the Flight

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Acknowledgments The author would like to express their sincere gratitude to researchers and engineers of JAXA, IHI and MHI in

this joint research and development works.

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557. 2Rosenmann, W., “Experimental Investigation of Hydrodynamically Induced Shaft Forces with a Three Bladed Inducer”,

Proceedings of the Symposium on Cavitation in Fluid Machinery, ASME Winter Annual Meeting, Nov.7-11, 1965, pp.172-195. 3Kamijo, K., Shimura, T. and Watanabe, M., “An Experimental Investigation of Cavitating Inducer Instability”, ASME paper

77-WA/FE-14, 1977. 4Kamijo, K., Yoshida, M. and Tsujimoto, Y., “Hydraulic and Mechanical Performance of LE-7 LOX Pump Inducer”, AIAA J.

Propulsion and Power, 9-6, 1992, pp.819-826. 5Uchiumi, M., Kamijo, K., Hirata, K., Konno, A., Hashimoto, T. and Kobayashi, S., “Improvement of Inlet Flow

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Inducer”, Journal of Propulsion and Power. AIAA, Vol.22, No1, 2006, pp.169-173. 7Tsujimoto, Y., Kamijo, K. and Brennen, C.E., “Unified Treatment of Flow Instabilities of Turbomachines”, Journal of

Propulsion and Power. AIAA, Vol.17, No.3, 2001, pp.636-643. 8Shimura, T., Yoshida, M., Kamijo, K., Uchiumi, M. and Yasutomi, Y., “A Rotating Stall Type Phenomenon Caused by

Cavitation in LE-7A LH2 Turbopump”. JSME International Journal (B), Vol.45, No.1, 2002, pp.41-46. 9Japikse, D., “Overview of Industrial and Rocket Turbopump Inducer Design”, Proceedings of the fourth International

Symposium on Cavitation (CAVI2001), SessionB7.001. 10Dorney, D., and Rothermel, J., “Simulations of Flow Through the SSME LH2 Feed Line and LPFP Inducer”, AIAA paper

2003-4421, 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Huntsville, Alabama, 20-23 July 2003. 11Rebattet, C., Wegner, M., Morel, P. and Bonhomme, C., “Inducer Design That Avoids Rotating Cavitation”, Proceedings of

the First International Symposium on Advanced Fluid Information, AFI-2001, Zao, Miyagi, 4-5 October 2001. 12Cervone, A., Testa, R., and d’ Agostino, L., “Thermal Effects on Cavitation Instabilities in Helical Inducers”, Journal of

Propulsion and Power. AIAA, Vol.21, No.5, 2005, pp.893-899.