liquid damping concentric membrane tank

8/2/2019 Liquid Damping Concentric Membrane Tank

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LIQUID DAMPING IN A CONCENTRIC MEMBRANE TANK

Keiji Komatsu, Professor, JAXA/Institute of Space and Astronautical Science,3-1-1 Yoshinodai, Sagamihara, 229-8510, Japan

Miki Nishimoto, Research Engineer, JAXA/Institute of Aerospace Technology,2-1-1 Sengen,Tukuba, 305-8505, Japan

ABSTRACTWe investigated the dynamic characteristics of the lateral and longitudinal (axi-symmetric) sloshing of liquid ina concentric membrane tank. We conducted an experiment using a circular cylindrical tank having a rigidouter wall and a flexible inner wall. The three materials used for the inner walls were, rigid acrylic, Kapton film,and polyethylene foam sheeting. We found that in the membrane wall, the outer part resonant phenomenonwas difficult to clearly discern, because of the heavy motion coupling of the inner and outer liquids through theflexible membrane wall. Consequently the apparent damping ratio was increased with decreasing rigidity ofthe membrane wall.

1. INTRODUCTION

Sloshing describes the free-surface oscillations of a liquid in a partially filled tank, which result from lateral andlongitudinal displacement or the angular motions of the vehicle. Our design configuration of aconcentric-membrane tank (Fig.1) should reduce the structural weight of a launch vehicle[1]. Due to thegeometric arrangement and flexibility of the inner tank, the attitude control and propulsion system stabilitymargins may change in comparison with the conventional tandem tank configuration[2].In the interaction between a liquid and a flexible membrane, we expect that the apparent damping in thesystem increases in the same manner as the flexible baffle in conventional tanks. This paper investigates thedynamic characteristics of liquid sloshing in a concentric tank.

2. MEASUREMENT SET-UP2.1 Model TankThe experiment tank was short cylindrical, with a rigid outer wall of 2R1=33.4-cmdiameter and an inner wall of2R2=23.4-cmdiameter(Figs.2-3 and Table 1).

2.2 Test Setup

In the experiment, we placed the tank on a base-excitation table (Figs.3-6). A mounted accelerometermonitored the base excitation frequency and its amplitude. A level meter and pressure gauge monitored theliquid motion.

Rigid Outer Tank

Flexible Inner Wall

R0

R1

R2

R3

H1

H0

R3

R2

R1

H1H2

Fig.1 Concentric membrane tank concept. Fig.2 Dimensions of the model tank.


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We attached the electrical-resistance level metervertically to the rigid outer wall for lateral excitationtest, and to the inner wall for vertical excitation test.The pressure gauge was on the bottom of the tank.We used three materials for the inner wall; rigidacrylic, Kapton film (0.1mm thick, Fig.4), andpolyethylene foam sheeting (1.8mm thick, Fig.6).Their elongation rigidity (Eh) ratio was : 1 : 0.003.Eh is one of the structural parameters thatdetermines the shell motion excited by the pressureof the sloshing liquid.

item dimensionInside diameter of the outer wall R1=16.7cmThickness of the outer wall R0-R1=6mmInside diameter of the inner tank R3=11.7cm

Outside diameter of the inner tank R2=12.5cmThickness of the inner acrylic rigidtank

R2-R3=8mm

Tank height H0=18cm

Water level of the outer part H1

Inner water level for inner acrylic rigidwall

H1

Inner water level for inner flexible wall H2

Thickness of the bottom plate H1-H2=1cm

Table 1 Dimensions of the model tank.

Anti-Aliasing Filter A/D Converter

Oscillator

Exciter

Level Gage

Amplifier

Amplifier

Accelerometer

Charge Amp.

Analysis

Fig.3 Diagram of response measurement Fig.4 Tank on the vertical excitation tablefor vertical excitation. ( Kapton inner wall case) .

Level Gage

Flexible wall

Analysis

Anti-Aliasing Filter - converter

Charge Amp.4ChargeAmp.2

PressureGage

Exciter

Accelerometer

Charge Amp.3

Amplifier

Oscillator

Fig.5 Diagram of response measurement Fig.6 Tank on the lateral excitation table

for lateral excitation. (foamed sheeting membrane case).

3. ANALYSIS

3.1 Sloshing Frequency in a Rigid TankThe fundamental sloshing frequency fin a rigid tank with radius Rand liquid level His given [3] as


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=

R

Htanh

R

g

2

1f

njnj (1)

with 841.111 = for lateral sloshing and

832.301 = for axi-symmetric (longitudinal) sloshing.

nj in equation (1) is

Rnj = (2)

where is obtained as a root of the equation0)R('J n = (3)

f (Hz) mode j=1 j=2

n method inner outer inner outer

Exact 0 0 2.85 4.00n=0BEM - - 2.87 4.14Exact 1.98 1.32 3.36 4.01N=1BEM 2.02 1.36 3.41 4.16Exact 2.56 1.87 3.77 4.05n=2BEM 2.61 1.95 3.82 4.20

Table 2 The accuracy of the eigen frequency forthe liquid elementH/R=0.6.

for a cylindrical tank, and for a concentric cylindrical tank

0)R('Y)R('J)R('Y)R('J 2n1n1n2n = (4)

where R1 is the outer diameter and R2 is the inner diameter. dx/)x(Jd)x(J . nJ and nY are the Besselfunctions of order nof first and second kind. nis a circumferential wave number andjis the mode number.

3.2 Coupled Oscillation ModelingWe applied a coupling method [4] for numericalcalculation. This method uses finite elements for atank and boundary elements for the liquid inside. Inmodeling, the geometry must be axi-symmetric, butthe motion may be non-symmetric because thecircumferential direction motions of the elements are

represented by a trigonometric function.

3.3 Boundary Element MeshFigure 7 illustrates tank-liquid meshing, where thelower part of the tank was filled with a liquid. Theliquid level was the same height in the inner part andthe outer part.

To estimate the accuracy of the meshed model, wecalculated the eigen frequencies for the rigid wallcase and compared them with exact theoreticalresults (by eqs (1)-(4)) in Table 2. To maintainaccuracy, we applied fine mesh around the freesurface.

4. TESTING PROCEDURE4.1 Lateral ExcitationWe used the following procedures to derive themodal parameters.

(1) The base was driven slowly to excite thefirst sloshing mode.

(2) We recorded the free decay time series(Fig. 8(a), the material of the inner wall inFig.8 was Kapton film). In this stage, the

17

18

19

32

31

30

29

28

27

16

15

14

13

9

8

7

6

5

4

3

2

10

23

24

22

LiquidLiquid

69

68

52 50 48 46 44 4251 49 47 45 43

11

12

1

53

41

40

39

38

37

36

35

34

33

72

73

74

75

21

Free Surface

Membrane Shell70

67 66 65 64 63 62 61 60 59 58 57 56 55 54

20

C.L.

17 18 19 20 21 22 23 24 25 26

Rigid Wall

71

Fig.7 Liquid-tank system element meshing

free decay data had several component modes. We transformed the time series into a frequencyspectrum (FRF, Frequency Response Function), and then identified the resonant frequency (Fig.8(b)), which was almost identical to the eigen frequency.

(3) The base was driven at the frequency identified by procedure (2).(4) At the resonant excitation, we suddenly stopped the driving force and again, recorded the free decay

data. We only had data for the first sloshing mode.(5) By counting the wave numbers and the time intervals of the free decay data, we determined the first

sloshing frequency. (Fig 8(c))

(6) Damping ratio was determined by the logarithmic decay ratio , calculated by the free-decaydata (Figs.8(c) and (d)) as

n

nN

a

alog

N

1,

2

+=

=

where na was the n-th amplitude peak of the free decay.

The water level is set at 5, 7.5, and 10cm, corresponded to H/R1 0.3, 0.45, 0.6, and H/R20.43, 0.64, 0.85.


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0 5 10 15 20 25 30 35 40 45-2

0

2

time

acc

0 5 10 15 20 25 30 35 40 45-2

0

2

time

pres

0 5 10 15 20 25 30 35 40 45-0.5

0

0.5

time

height

0 1 2 3 4 50

20

40

60

80

100

Hz

P/A

0 1 2 3 4 50

50

100

150

Hz

H/A

Fig.8(a) Impulse response for rigid wall. Fig.8(b) FRFs for Fig.9(b).

Inner part is measured by a pressure gage, outer part by a wave gage.

35 40 45 50 55 60 65 7 0 75 80-0.2

0

0.2

time

acc

35 40 45 50 55 60 65 7 0 75 80-1

0

1

time

pres

35 40 45 50 55 60 65 7 0 75 80-0.5

0

0.5

time

height

10 15 20 25 30 35 40-0.1

0

0.1

time

acc

10 15 20 25 30 35 40-0.5

0

0.5

time

pres

10 15 20 25 30 35 40-0.01

0

0.01

time

height

Fig.8(c) Free decay record after the resonance Fig.8(d) Free decay record after the resonance

of the outer part. of the inner part.

Figure 9 presents the data for the foam wall. In the frequency spectrum (Fig.9(b)) we could not find a distinctresonant frequency for the outer part. The inner and outer parts interacted closely through the flexible wall,

and in the inner and outer parts, resonant phenomena were hardly excited. Figure 10 depicts the vibrationmode of the membrane wall. Its circumferential wave number was n=6. In linear theory, the excitation is alateral sloshing mode, therefore the n=1 mode must be observed. In reality, however, the inner wall resonatedwith the sloshing frequency of the n=6breathing vibration mode, whose frequency approximately equaled thesloshing frequency.We observed that in the membrane wall, the outer resonant phenomena was difficult to clearly discern,because of the heavy motion coupling of the inner and outer liquids through the flexible membrane wall.

0 5 10 15 20 25 30-0.5

0

0.5

time

acc

0 5 10 15 20 25 30-1

0

1

time

pres

0 5 10 15 20 25 30-0.2

0

0.2

time

height

0 1 2 3 4 50

20

40

60

80

100

Hz

P/A

0 1 2 3 4 50

10

20

30

Hz

H/A

Fig.9(a) Impulse response for foam wall. Fig.9(b) FRFs for Fig.10(b).

Inner part is measured by a pressure gage, outer part by a level gage.


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30 3 5 40 45 50 55-0.1

0

0.1

time

acc

30 3 5 40 45 50 55

-0.2

0

0.2

time

pres

30 3 5 40 45 50 55-0.1

0

0.1

time

height

30 35 40 45 50 55 60-0.1

0

0.1

time

acc

30 35 40 45 50 55 60-0.5

0

0.5

time

pres

30 35 40 45 50 55 60

0

0.02

0.04

0.06

time

height

Fig.9(c) Free decay record Fig.9(d) Free decay record

after the resonance of the outer part. after the resonance of the inner part.

4.2 Vertical ExcitationFor the vertical excitation, the wave motion frequency is exactly one-half that of the excitation. This type ofmotion is known as one-half sub-harmonic response[5].In the test procedure stated in 4.1, Steps (1)-(3) were excluded, because we had no way to give the tank

axi-symmetric impulse. Figure 11 present the excitation acceleration and the liquid level response. As can beseen, the frequency of the liquid motion is exactly one-half of that of the base acceleration.

Fig.10 Breathing vibration mode (n=6) for theflexible wall

Kapton FoamYoungsmodulus, E

3109

kg/ms2

5.83105

kg/ms2

Thickness,h 0.1mm 1.8mmDensity, 1410kg/m3,

(1.42)27.1kg/m

3,

(0.027)Elongation

rigidity, Eh310

510.510

2

Bendingrigidity Eh

3310

-33.410

-3

Table 3 Elastic modulus of the membrane sheets

0 2 4 6 8 10 12-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

Magnitude

Input acceleration

0 2 4 6 8 10 12-0.1

-0.05

0

0.05

0.1

0.15

Time (sec)

Magnitude

Free surface response

0 2 4 6 8 10 12 140

50

100

150

200

Frequency (Hz)

Magnitude

Input acceleration

0 2 4 6 8 10 12 140

5

10

15

20

25

30

Frequency (Hz)

Magnitude

Free surface response

Foam sheet : H=5.0cm (Upper: Input acceleration, Lower: Free surface motion)

Fig.11(a) One-half sub-harmonic liquid motion response Fig.11(b) Input and Output Frequency Spectraunder vertical excitation.


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5. RESULTS AND DISCUSSIONS

5.1 Damping Characteristics (Lateral Sloshing)Figures 12 and 13 present the measured equivalent damping ratios. We determined them by the logarithmicdecay method. In these figures, (a) represents the low amplitude case, and (b) represents the high amplitudecase.In Fig. 12, we calculated the damping ratio using the free-decay records of the level meter attached to the wallof the outer tank. In Fig. 13, we calculated the damping ratio using the record of the pressure gauge placed atthe bottom of the inner tank.

The definition of high amplitude is that the wave height is not so high that it will prevent the water particlesfrom separating from the free surface. We used ten waves to calculate the damping ratio. The definition oflow amplitude is that the wave height is not so sufficient to maintain a precise signal to noise ratio. The rigidityof the membranes are estimated in Table 3.

0

1

2

3

4

5

6

5cm 7.5cm 10cm Level

Dampingratio(%)

rigid

Kapton

foam

0

1

2

3

4

5

6

5cm 7.5cm 10cm Level

Dampingratio(%)

rigid

kapton

foam

(a) Small amplitude excitation. (b) Large amplitude excitation.

Fig.12 Damping ratios for the outer part [Lateral sloshing]

0

1

2

3

4

5

6

5cm 7.5cm 10cmLevel

Dampingratio(%)

rigid

Kapton

foam

0

1

2

3

4

5

6

5cm 7.5cm 10cmLevel

Dampingratio(%)

rigid

kapton

foam

(a) Small amplitude excitation. (b) Large amplitude excitation

Fig.13 Damping ratios for the inner part [ Lateral sloshing]

0

0.5

1

1.5

2

2.5

3

5cm 7.5cm 10cm

Level

Dampingratio(%)

rigid

Kapton

form

0

0.5

1

1.5

2

2.5

3

5cm 7.5cm 10cm

Level

D

ampingratio(%)

rigid

Kapton

form

(a) Small amplitude excitation. (b) Large amplitude excitation

Fig.14 Damping ratios for the inner part. [Axi-symmetric sloshing]

As can be seen in the figures, the damping ratio increased with decreasing membrane wall rigidity.


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5.2 Damping Characteristics (Axi-symmetric Sloshing)Figure 14 presents the measured equivalent damping ratios for vertical excitation by using the wave heightrecords. The resonant axi-symmetric mode was observed clearly for the inner part, however, the outer partresonant mode was not always axi-symmetric, but non-symmetric mode (n was 4-10, depending on theexcitation frequency, shown in Fig.17.).

5.3 Comparing Measured and Calculated FrequenciesSection 3.2 of this paper explains the combined methods used to calculate eigen frequencies and eigenmodes. For the input data of the foam, we measured the bending deflection caused by its own weight and we

applied a tensile test to the specimen (Table 3).

Liquid level H/R1=0.3(5cm) H/R1=0.45(7.5cm) H/R1=0.6(10cm)WallRigidity Eh Outer Inner Outer Inner Outer InnerExperiment 0.78 1.64 0.95 1.83 1.05 1.95Exact 0.77 1.60 0.92 1.80 1.03 1.89

Rigid(R2=11.7cm)

BEM

0.95 1.67 1.14 1.94Experiment 0.78 1.50 0.94 1.76 1.04 1.90Kapton

(R2=12.5cm) FEM-BEM3.010

5kg/m

2

0.95 1.67 1.14 1.93Experiment (0.74*) 1.45 (0.75*) 1.65 (0.88*) 1.65Polyethylene

Foam FEM-BEM1.010

2kg/m

2

0.65 1.39

Table 4 Coupled eigen frequencies* resonance is not observed clearly [Lateral sloshing].

Lateral Excitation (H/R=0.3)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1 2 3 4 5

Rigidity of the wall -log(E/E0)

Frequency(Hz)

Outer 1st

Inner 1st

Inner 2nd

Outer 2nd

Fig. 15 Calculated lateral sloshing frequencies as a function of membrane rigidity.

Tables 4 and 5 compare measured and calculated frequencies. Figures 15 and 16 illustrate the relationshipbetween the Youngs modulus and the eigen frequencies of the system. Calculated eigen modes are shown inFigs. 18 and 19 (corresponding to Fig.7). As seen in the figures and tables, reducing the inner wall rigiditydoes not significantly decrease the resonant frequency. The important characteristic of the flexible tank is thatthe lowest resonant frequency can be calculated, however, its resonant phenomena is hardly observedobviously. The high damping ratio is a result of this phenomenon, which can only be observed by theexperiment.

Liquid level H/R1=0.3(5cm) H/R1=0.45(7.5cm

)

H/R1=0.6(10cm)Wall

Rigidity Eh Inner Inner InnerExperiment 2.72 2.82 2.88Exact 2.75 2.83 2.85

Rigid(R2=11.7cm)

BEM

2.75 2.86Experiment 2.60 2.80 2.85Kapton

(R2=12.5cm) FEM-BEM3.010

5kg/m

2

2.74Experiment 2.65 2.80 2.86Polyethylene

Foam FEM-BEM1.010

2kg/m

2

2.71

Table 5 Coupled eigen frequencies* resonance is not observed clearly[Axi-symmetric sloshing]

Exp.(Kapton)Exp.(Foam)


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Vertical Excitation (H/R=0.3)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1 2 3 4 5Rigidity of the wall -log(E/E0)

Frequency(Hz)

Inner 1st

Inner 2nd

Outer 1st

Fig. 16 Calculated axisymmetric sloshing frequencies as a function of membrane rigidity

Fig.17 Response of the outer part Fig.18 Calculated first lateral sloshing modeunder vertical excitation. (H/R=0.3, inner wall : -log(E/E0)=3.4)

Fig.19 Calculated first and second axi-symmetric sloshing modes(H/R=0.3, inner wall : -log(E/E0)=3.4)

6. CONCLUSIONSWe found that when the inner wall was flexible, the amplitude of the resonant frequency was low. It wasdifficult to find this resonant phenomenon and the damping ratio increased with decreasing membrane wallrigidity.

References

[1] Komatsu,K.,Sano,M.,Kimura,J., and Ohyagi,T. : Concept Study of Membrane Tanks for Launch Vehicle,AIAA/ASME/ASCE/AHS/ASC, 41st Structures, Structural Dynamics, and Material Conf., Atlanta, 2000, Apr.5,.[2] Nikolayev,O. and Komatsu,K. : Propulsion System Instability for Concentric Tank-Type Launch Vehicle,AIAA J. of Propulsion and Power, Vol.20, No.2, pp.376-378,2003.[3] Bauer,H.F. : Fluid Oscillations in the Containers of a Space Vehicle and Their Influence upon Stability,NASA Technical Report, NASA TR R-187,1964.[4] Komatsu,K. and Matsushima,M. : Some Experiments on the Vibration of Hemispherical Shells PartiallyFilled with a Liquid, J. of Sound & Vibration, pp.35-44 ,64-1 ,1979.[5] Dodge,F.T. : Vertical Excitation of Propellant Tanks, Chapter 8 in The Dynamic Behavior of Liquid inMoving Containers, NASA SP-106,1966.

Exp.(Kapton) Exp.(Foam)

Free surface motion

Membrane deformationCenterline of the tank

liquid damping concentric membrane tank

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