The Application ofFEM-BEM Approach

to Simultaneous Dynamic Analysisof Several Thin-walled Fuel Tanks

Robert Lamper ,Vladimir Levin, Nicolay Pustovoy

Novosibirsk State Technical UniversityK. Marx pr., 20, Novosibirsk, 630092, Russia,Phone: +73463 121, Fax: +73460209

Abstract - At research of dynamics of longitudinal movement ofthe flying device with big weights of fuel the problem aboutfluctuations of thin-walled fuel tanks preliminary is solved. Insuch problems has well proved a method of finite and boundaryelements. This approach allows to lead direct calculation of asheaf of tanks, that considerably facilitates procedure ofconstruction of mechanical analogue. Dynamic pressure of aliquid upon stacks of a tank within the limits of a direct method ofboundary elements is expressed through movings shells and themass and rigidity matrix of the attached weights of a liquid whichjoins in the settlement scheme together with matrixes of weightsand a tank, received as a result of digitization of a design of a tankon FEM is calculated. After the decision of the generalizedproblem of own values for the received matrixes integratedcharacteristics of fluctuations which can be used at constructionof mechanical or "matrix" analogue of a tank for the decision ofthe general problems of flying device dynamics can be found. Theway of the description of geometry of a tank on the basis ofnatural approximation of a curve is offered.

I. INTRODUCTION

The problem of perfection existing and development of newcomputational techniques of dynamics of thin-walled fuel tanksremains actual. Especially sharply it is shown at designingdesigns of aviation and rocket engineering.For the solution of problems of dynamics of longitudinal

motion of launch vehicles of space vehicles with liquid rocketengines the special mathematical model - the so-calleddynamic scheme of a design, as a rule, is used. The importantstage in its drawing up is in-depth study of dynamic propertiesof the separate fuel tanks including in the common design.With this purpose the problem about own longitudinaloscillations of each thin-walled fuel tank with a liquid issolved. Above the solution of this challenge many researchersworked. Computational methods developed together withdevelopment of computer facilities. Recently to similarproblems the methods based on certainly-elementrepresentation of a design are applied. In particular, in suchproblems has well proved a method based on a combination ofa method of final elements for elastic elements of a tank andboundary elements for a liquid being a tank [4].

After the solution of a problem on own oscillations of a tankits

Fig. 1. The computational scheme of a tank

"mechanical" [5] analogue representing a set of weights,connected by springs, or the "mathematical" analoguerepresenting corresponding matrixes in which dynamicproperties of a tank (for example are considered is usuallymade, dynamic condensation) is conducted. The analoguereproduces the peak-frequency characteristic of a tank in thedetermined frequency band.New general-arrangement diagrams and constructive

solutions sometimes do not allow to construct mechanicalanalogue of a sheaf of tanks in the conventional image.Attempts of realization of such approach are interfaced to thedetermined difficulties [1-4]. One of exits is direct calculationof several tanks simultaneously. The algorithm based oncombination FEM and BEM, allows to conduct suchcalculation. Initial fillings up of consistently disposed tanks canbe such, that the bottom of the upper tank is moistened with aliquid of the lower tank. Algorithm we shall apply and in thisdifficult case.Here the example of the direct analysis of dynamics of a

sheaf of two cylindrical tanks with the hemispherical bottomsis considered.

II. PROPOSED METHOD

The thin-walled elastic tank partially filled by a liquid madeof shells of revolution (fig. 1) is considered. At definition offrequencies and forms of longitudinal (axisymmetrical)oscillations of an elastic tank with a liquid known assumptions

receive. For thelZ approximate description

of a tank with a liquid assystems with infinitenumber of degrees offreedom system with finalnumber of degrees offreedom it is used twoknown methods - amethod of final elements

|V S \ for representation of

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elastic elements of a tank and a method of boundary elementsfor modelling motions of a liquid in a tank [4]. The contour ofa tank (meridian) is set by coordinates of discrete points anddirections of external normals to a contour in these points.Finite elements of the moistened surface of a tanksimultaneously are boundary elements for a volume of a liquid.The liquid filling a tank receives incompressible and

nonviscous. Its motions are described by potential ofmovements. For the approximate description of effect of aliquid on a tank at axisymmetrical oscillations the approachbased on a direct method of boundary elements is used.Application of this approach allows to write down function ofkinetic energy of a liquid in the form of the quadratic form ofthe generalized nodal unknown persons of the certainly-element scheme

The problem about own oscillations of a tank with a liquid isformulated as the generalized problem of own values. Thesolution is conducted in two stages. At the first stage within thelimits of a direct method of boundary elements the matrix ofthe attached weights of a liquid is calculated. At the secondstage within the limits of a method of final elements the matrixof weights and a matrix of rigidity of a dry tank are calculated.Further to a matrix of weights of a dry tank the matrix ofweights of a liquid and a problem is added is reduced to thegeneralized problem of own values for the general matrixes ofrigidity and weights.

After the solution of the primary goal of a presence offrequencies of own oscillations and corresponding own vectorsof certainly-dimensional model the problem of graphicrepresentation of each vibration mode of a tank can be solved:movements of a shell and pressure profile. If necessary byresults of the solution of a problem on own values themechanical analogue of a tank which with the determinedaccuracy replaces a tank in the general problem of dynamiccalculation of a design of the launcher [4] can be constructed.

For definition of parameters of analogue on each vibrationmode and the control of correctness of calculation integralcharacteristics are calculated. They can be used and for thecontrol of correctness of calculations. So, for example, on eachvibration mode the total longitudinal momentum should beequal to the total longitudinal force perceived by referencebulkheads.

As dynamic effect of a liquid on walls of a tank is expressedthrough movements of a wall, the description of acomputational case of a tank with the intermediate bottom, andsimplly sheaf of several tanks is possible. In some casesconstruction of mechanical analogue of a sheaf is complicated,and sometimes and it is simplly impossible (for example, for asheaf of toro-cylindrical tanks)

II. RESULTS OF ANALYSIS

As an example of calculation the sheaf of two identicalcylindrical tanks with the hemispherical bottoms (fig.2.) wasconsidered. The length of the cylindrical lip of each tank is

adopted equal to three radiuses of the bottom. Fuel tanks canbe actuated in the computational scheme on a miscellaneous.For corresponding version calculation on own oscillations byrespective image of the fixed tanks is conducted. On fig. 2 it isshown theoretically possible three versions of longitudinalfastening of a sheaf of tanks:

a) On the upper bulkheadb) On the upper and lower bulkheadsc) On three bulkheads simultaneouslyIn the latter case, - to an essence, it is conducted

simultaneous calculation of two identical tanks, therefore at

-

Tank

Tank 2

a) b) c)

Fig. 2. The scheme with a consecutive arrangement of cylindricaltanks. Kinds of fastenings in a longitudinal direction

absence of hydrodynamic interaction of tanks (for incompletefillings up of the lower tank) there is a capability of the internalcontrol of algorithm of calculation of a sheaf of tanks.Levels of a filling up of tanks h, h2 are read out from poleseach tank. Parameters of the computational scheme thefollowing: Material of shells of tanks

aluminium, pg = 1000 A3 density of a liquid,

B = 3.846 10' H characteristic rigidity, R = m - radius of a

tank.The sheaf of tanks can be actuated in a design of the launcher

by means of three bulkheads on versions a), b), c) (fig.2). Bymeans of standard techniques for calculation the case is mostsimple. Here calculation of each tank separately thenparameters of mechanical analogue of each tank aredetermined is conducted. In version of fastening a referencebulkhead one. Theoretically, to the lower bulkhead of thesecond tank the engine can fasten. Construction of analogue incase of comes across the determined difficulties. DevelopedFEM-BEM the algorithm does not meet any difficulties atcalculation of a sheaf of tanks without hydrodynamic and withhydrodynamic connection (a tank with the intermediate

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bottom, in this case at h2> 3R ). Results of calculations of firstten frequencies of own longitudinal oscillations (Hz) a sheaf oftanks are resulted in table 1.

TABLE 1FREQUENCIES OF OWN OSCILLATIONS, HERTZ

a) b) c)

33.63 57.64 72.7861.46 72.78 73.11

n =2R 101.0 105.6 112.2105.8 112.3 113.7

h2 =2R 149.2 152.1 161.6152.5 161.6 161.6181.0 181.1 181.2183.2 183.2 183.3207.5 209.3 210.7209.5 210.7 211.529.44 47.2 55.0949.87 55.09 55.3484.61 94.19 109.8

k = 2.5R 95.22 109.9 110.8132.6 134.1 137.3

h2= 2.5R 134.1 137.3 137.9164.2 166.7 169.5167.0 169.5 170.4188.1 189.4 189.4189.6 189.6 190.4

24.80 36.41 38.9653.98 58.16 73.1269.25 95.76 96.13

&= 2.R 104.4 106.0 112.3108.9 112.3 113.7

h2= 3.5R 137.9 140.8 140.8151.8 152.4 161.6160.3 166.5 166.6179.1 179.1 179.1183.2 183.2 183.3

In case of equal small fillings up of tanks calculationsimultaneous calculation of two identical tanks, in essence, isconducted. In table 1 it is a case c).In the computational scheme of a tank angular rigidity of asupport is not modelled, therefore in pairs frequencies there is asmall difference. Theoretically these pairs frequencies shouldbe identical. On fig 3,4 first two forms of own oscillations of asheaf of tanks are resulted at various fastenings and levels of afilling up.

AZ

1 24.80 Hz2J 53.98 Hz3) 69.25 Hz4] 104.4 Hz5] 108.9 Hz6I 137.9 Hz71 51.8 Hz8) 160.3 Hz9) 179.1 Hz101 183.2 Hz

El 53.98 Hz (2)

* 24.80 Hz (1J

Fig. 4. First two forms of own oscillationsof a sheaf of tanks

IV. CONCLUSIONS

Az

1) 33.63 Hz21 61.46 Hz31 1 01.0 Hz4) 105.8 Hz51 149.2 Hz61 152.5 Hz71 181.0 Hz81 183.2 Hz91 207.5 Hz

1 0) 209.5 Hz

061.46 Hz (2J33.63 Hz 111

Fig. 3. First two forms ofown oscillationsof a sheaf of tanks

In this article the feasibility before developed algorithm FEM-BEM of calculation of own longitudinal oscillations of a sheafof thin-walled fuel tanks, including in a difficult case ofhydrodynamic interaction is illustrated.The article is written at support of the analytical

departmental target program << Development of scientificpotential of the higher school 2006-2008 >> PHII 2.1.2.2676.

REFERENCES

[1] Kolesnikov K.S. "Longitudinal oscillation of a rocket with liquid rocketengine".-M: Engineering, 1971. 270pp.[2] Lamper R.E., Levin V.E. "To definition of parameters of analogue of a fueltank by results of its calculation by a method boundary and finite element". HScientific bulletin NSTU.- 1 999.Xo2 (7) - P. 171 - 176.[3] Lamper R.E., Mileev D.D., Teslenko A.A. "About mechanical analogue forlongitudinal oscillations of an axisymmetrical elastic tank" H Oscillations ofelastic designs with a liquidl The collection ofproceedings of a symposium.-Novosibirsk.-1973. P. 115-123.[4] Lamper R.E., Levin V.E. "Method of finite and boundary elements inproblems of dynamics of elastic vessels with a liquid". PMMI, Xol 2004., P.93-97[5] Pogjalostin A.A. "Definition of parameters of mechanical analogue foraxisymmetrical oscillations of an elastic cylindrical vessel with a liquid". H Theengineering log-book ofthe Mechanic ofafirm body.-1966. P.157-159.

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