JP5644459B2 - Low gravity sloshing damping ratio prediction method and low gravity sloshing design model creation method - Google Patents

Description

The present invention is low weight Chikarasu used to predict the mode damping ratio of sloshing that is required when performing analysis of the sloshing that occurs in the storage liquid in the tank which is arranged in a low gravitational field, such as the tank of the spacecraft The present invention relates to a lossy damping ratio prediction method and a low-gravity sloshing design model creation method using the damping ratio prediction method.

  In general, when a tank structure is designed for a tank that stores liquid, it is essential to accurately predict a load due to sloshing dynamic pressure accompanying sloshing of the liquid (storage liquid) generated in the tank. Therefore, prediction of the viscous damping ratio is required for the sloshing of the liquid in the tank as described above.

  By the way, in a liquid storage tank such as a propellant tank mounted on a spacecraft such as a space satellite, if the sloshing of the liquid occurs in the tank, the attitude control of the spacecraft itself may be affected.

  Therefore, even in a liquid storage tank mounted on this type of spacecraft, it is required to predict a viscous damping ratio for sloshing of the stored liquid.

  In the tank provided in the ground gravity field, since the surface tension acting on the liquid stored in the tank is significantly smaller than the gravity acting on the liquid, the liquid level of the liquid in the tank is It is almost flat, and a meniscus is formed only slightly on the outer peripheral edge due to the influence of surface tension. Therefore, when the damping ratio of the sloshing of the liquid in the tank in the ground gravity field is predicted, the surface tension of the liquid can be almost ignored.

  However, since the outer space is a low-gravity field, the liquid stored in the spacecraft tank has a greater influence of surface tension and a larger meniscus as the influence of gravity decreases. . Therefore, for liquids stored in tanks that are placed in a low-gravity field such as a spacecraft tank, it is important to predict the viscous damping ratio for low-gravity sloshing considering the surface tension of the liquid. .

  In practice, there are many cases where the fundamental mode of the liquid resonates with respect to the tank excitation input, and under such a resonance, it is possible to accurately know the mode damping ratio of the low-gravity sloshing of the liquid that occurs in the tank. It is necessary to calculate the sloshing wave height due to the low gravity sloshing of the liquid in the tank, the force acting on the tank, and the moment.

  As for the low-gravity sloshing in the tank placed in the low-gravity field as described above, it has been said that it is difficult to apply the analytical method even in the case of non-viscous sloshing in the general axisymmetric tank other than the cylindrical tank. The present inventor has proposed a method in which a velocity potential and a liquid level displacement satisfying the Laplace equation can be expressed analytically for a general axisymmetric tank by introducing appropriate spherical coordinates. (For example, refer nonpatent literature 1).

  In addition, a low-gravity sloshing has also been proposed as a method for creating a mass spring design model when damping is not included (see, for example, Non-Patent Document 2).

  Furthermore, several empirical formulas have been reported by experimental studies on the damping ratio of the modeled low gravity sloshing in a spherical tank and a cylindrical tank having an ellipsoidal bottom (see Non-Patent Document 3, for example). ). The modeling in Non-Patent Document 3 is not actually an experiment in a low-gravity space, but even on the ground, the tank size can be reduced to increase the effect of surface tension relative to gravity. This means that we have modeled a low-gravity environment where the effect of surface tension becomes stronger using a small tank on the surface.

Utsumi, M., “Low-gravity Sloshing in an Axisymmetrical Container Excited in the Axial Direction”, American Society of Mechanical Engineers (ASME) , Journal of Applied Mechanics, Vol. 67, June 2000, pp. 344-354 Utsumi, “A Mechanical Model for Low-Gravity Sloshing in an Axisymmetric Tank”, American Society of Mechanical Engineers (ASME), Journal of Applied Mechanics (Journal of Applied Mechanics), Vol. 71, September, 2004, pp. 724-730 Dodge, FT, Garza, LR, “Simulated Low-Gravity Sloshing in Spherical Tanks and Cylindrical Tanks with Inverted Ellipsoidal Bottoms) ”, National Aeronautics and Space Administration (NASA), Technical Report No. 6 (Technical Report No. 6), Contract No. NAS8-20290 (Contract NAS8-20290), 1968, pp. 196 1-34

  However, as shown in Non-Patent Document 3 described above, the method of conducting experimental research on low gravity sloshing of the liquid in the tank is inconvenient because it requires a lot of time and cost for parameter studies. .

  Note that Non-Patent Document 1 and Non-Patent Document 2 do not particularly mention prediction of the viscous damping ratio of low-gravity sloshing of the liquid in the tank in which surface tension is important in a low-gravity field, and also relates to low-gravity sloshing. The mass spring design model including damping due to the viscosity of the liquid is not particularly mentioned.

  Therefore, the present invention theoretically calculates the damping ratio when the sloshing generated in the liquid stored in a tank arranged in a low-gravity field such as a spacecraft tank is generated based on the viscosity of the liquid. And a low-gravity sloshing damping ratio prediction method used to enable easy prediction, and a design model corresponding to a low-gravity sloshing system having a damping ratio predicted by the damping ratio prediction method. It is intended to provide a design model creation method of low gravity sloshing so that it can be created.

In order to solve the above-mentioned problems, the present invention, corresponding to claim 1, represents a general axisymmetric tank arranged in a low-gravity field in spherical coordinates, and the liquid inside the general axisymmetric tank is non-viscous. Assuming that, the mode equation of non-viscous sloshing based on the variational principle including the surface tension of non-viscous fluid is obtained, and then the virtual term in the viscous boundary layer formed on the tank wall surface is obtained by the viscosity term of the Navier-Stokes equation. The work is represented by mode coordinates, and the virtual work in the viscous boundary layer formed on the tank wall surface represented by the mode coordinates is added to the mode equation of the non-viscous sloshing, and the time of the mode coordinates corresponding to the velocity term A method for predicting the damping ratio of low gravity sloshing to obtain a mode equation including a first-order differential term.

Further, according to claim 2, the general axisymmetric tank disposed in the low gravity field is represented by spherical coordinates, and the surface tension of the non-viscous fluid is assumed assuming that the liquid inside the general axisymmetric tank is non-viscous. Finds the mode equation of non-viscous sloshing based on the variational principle including, and then expresses the virtual work in the viscous boundary layer formed on the tank wall surface by the mode coordinate in the viscosity term of the Navier-Stokes equation. In addition to the non-viscous sloshing mode equation, the mode equation including the first-order differential term according to the time of the mode coordinate corresponding to the velocity term is obtained. Next, the sloshing system in the case where sloshing occurs in the horizontal uniaxial direction of the liquid inside the general axisymmetric tank using the mode coordinates obtained as the solution of the mode equation will be described. The uniaxial force acting on the general axisymmetric tank along with the sloshing and the moment about the axis orthogonal to the uniaxial direction in a horizontal plane are obtained, and the uniaxial force and the axis orthogonal to the sloshing system are obtained. The response to the vibration acceleration of the surrounding moment is obtained, while the fixed mass is fixed at a certain height position in the tank and the movable mass arranged at another certain height position is placed on the tank wall surface as a spring element. And a temporary design model configured to be supported via a damper, and based on the equation of motion of the movable mass in the design model, the uniaxial force acting on the tank of the temporary design model, and In addition to obtaining the moment around the orthogonal axis, the force in the uniaxial direction in the design model and the same acceleration as the above for the moment around the orthogonal axis The response is obtained, and then the uniaxial force in the sloshing system and the response to the excitation acceleration of the moment about the axis orthogonal thereto, the uniaxial force in the temporary design model and the moment about the axis orthogonal thereto The damping coefficient required under the condition that the response to the same excitation acceleration of the above is equal for any excitation frequency is obtained, and further, the uniaxial force in the sloshing system and the axis around the axis orthogonal thereto are obtained. Correction necessary to correct the difference between the response of the moment to the excitation acceleration , the force in the uniaxial direction in the temporary design model, and the response to the same excitation acceleration of the moment about the axis orthogonal to the moment Obtain the force and correction moment, and then move the movable mass into the tank via the uniaxial spring element and damper. Low gravity sloshing design that creates a design model by determining the mass of the movable mass and its mounting position so that the force and moment generated by the damper are equivalent to the correction force and correction moment. Model creation method.

According to the present invention, the following excellent effects are exhibited.
(1) Based on the variation principle including the surface tension of a non-viscous fluid assuming that the general axisymmetric tank placed in a low-gravity field is expressed in spherical coordinates and the liquid inside the general axisymmetric tank is non-viscous. The mode equation of non-viscous sloshing is obtained, and then the virtual work in the viscous boundary layer formed on the tank wall surface is expressed in the mode coordinate by the viscosity term of the Navier-Stokes equation, and the tank wall surface expressed by the mode coordinate the virtual work in a viscous boundary layer that can be, in addition to the mode equations above inviscid sloshing, low gravity sloshing that way determine the mode equations including first-order differential term due to the time mode coordinates corresponding to the speed terms Since it is a damping ratio prediction method, in addition to the above-mentioned non-viscous sloshing mode equation derived from the variational principle considering the surface tension of the liquid, the time of the mode coordinate corresponding to the velocity term Based on the mode equations including first-order differential term with, it is possible to perform the damping analysis of reduced gravity sloshing in consideration of the surface tension of the liquid.
(2) Therefore, the damping ratio based on the viscosity of the low gravity sloshing generated in the liquid stored in the general axisymmetric tank disposed in the low gravity field can be theoretically obtained. For this reason, the analysis for obtaining the damping ratio of the low gravity sloshing can be greatly facilitated as compared with the conventional method of performing experimental research.
(3) In the same procedure as in (1) above, in addition to the mode equation of inviscid sloshing, a mode equation including a first-order differential term depending on the time of the mode coordinate corresponding to the velocity term is obtained, and then the solution of the mode equation is performed. With respect to the sloshing system in which sloshing occurs in the horizontal uniaxial direction of the liquid inside the general axisymmetric tank using the mode coordinates obtained as follows, the uniaxial direction acting on the general axisymmetric tank along with the sloshing And the response around the axis perpendicular to the uniaxial direction in the horizontal plane, and the response to the excitation acceleration of the uniaxial force and the moment around the axis orthogonal to the sloshing system, A fixed mass is fixed at a certain height position and a movable mass arranged at another certain height position is placed on the tank wall with a spring element and A temporary design model configured to be supported via a pad is set, and from the equation of motion of the movable mass in the design model, the uniaxial force acting on the tank of the temporary design model and the orthogonal And a response to the same excitation acceleration of the uniaxial force in the design model and the moment around the axis orthogonal thereto, and then the uniaxial force in the sloshing system The response to the excitation acceleration of the moment around the axis perpendicular to it, and the response to the same excitation acceleration of the force in the uniaxial direction and the moment around the axis perpendicular to the uniaxial force in the temporary design model are arbitrary excitation frequencies. To obtain the damping coefficient required under the condition that is equal to each other, and further, the uniaxial force in the sloshing system Corrects the difference between the response to the excitation acceleration of the moment around the axis orthogonal to it, the force in the uniaxial direction in the temporary design model, and the response to the same excitation acceleration of the moment about the axis orthogonal to it The correction force and the correction moment required for the operation are obtained, and then the movable mass is mounted in the tank via the uniaxial spring element and the damper, and the force and the moment caused by the damper are corrected as described above. The low-gravity sloshing design model creation method that creates the design model by determining the mass of the movable mass and its mounting position so that it is equivalent to the force and the correction moment. Is adopted as the damping ratio of the design model as it is, and the design model based on the conventional general idea is modified and placed in the low gravity field. A low-gravity sloshing design model capable of satisfactorily giving a damping ratio due to the viscosity of the low-gravity sloshing generated in the liquid in the tank can be constructed.

It is a figure which shows the spherical coordinate of this tank in the case of applying to a spherical tank as one Embodiment of the attenuation ratio prediction method of the low gravity sloshing of this invention. It is a figure which shows the change by the liquid filling level (liquid filling level) of the tank of the mode damping ratio (Damping ratio) determined by the damping ratio prediction method of FIG. It is a figure which shows the change by the liquid filling level (liquid filling level) of the tank of the damping coefficient (Damping coefficient) derived | led-out from the mode damping ratio determined by the damping ratio prediction method of FIG. It is a figure which shows the change by the liquid filling rate (liquid filling level) of the tank of the mass parameter (Mass parameter) derived | led-out from the mode damping ratio determined by the damping ratio prediction method of FIG. It is a figure which shows the change by the liquid filling level (liquid filling level) of the tank of the value which remove | divided the damping coefficient derived from the mode damping ratio determined by the damping ratio prediction method of FIG. 1 by the mass parameter. It is a figure which shows the change by the liquid filling rate (liquid filling level) of the tank of the natural period derived | led-out from the mode damping ratio determined by the damping ratio prediction method of FIG. The figure which shows the change by the liquid filling rate (liquid filling level) of the tank of the radius of the contact line with respect to the tank wall surface of a liquid level used in order to verify the effectiveness of the mode damping ratio determined by the damping ratio prediction method of FIG. It is. It is a figure which shows the change by the liquid filling level (liquid filling level) of the tank of a correlation parameter used in order to verify the effectiveness of the mode damping ratio determined by the damping ratio prediction method of FIG. It is a figure which shows the change by the liquid filling rate (liquid filling level) of the tank of the meniscus used in order to verify the effectiveness of the mode damping ratio determined by the damping ratio prediction method of FIG. In order to verify the effectiveness of the mode damping ratio determined by the damping ratio prediction method of FIG. 1, a comparison with the experimental results of Non-Patent Document 3 is shown. (A) and (b) are the conditions under which the parameters are changed. It is a figure which shows the result in. It is a figure which shows the design model temporarily set in the process of implementing the design model creation method of low gravity sloshing as another form of implementation of this invention. It is a figure which shows the design model created with the design model creation method of the low gravity sloshing of this invention.

  Hereinafter, embodiments for carrying out the present invention will be described with reference to the drawings.

  FIGS. 1 to 10 (a) and 10 (b) show an embodiment of the low-gravity sloshing damping ratio prediction method according to the present invention, for example, as shown in FIG. 1 as a general axisymmetric tank arranged in a low-gravity field. A case where the present invention is applied to the prediction of the damping ratio of the low gravity sloshing of the liquid stored in the tank in the spherical tank 1 is as follows.

  That is, in general, the liquid is viscous, but in order to perform the attenuation analysis of the low gravity sloshing considering the surface tension of the liquid, the non-viscous sloshing for a complete fluid (non-viscous fluid) in which the liquid is assumed to be non-viscous in advance. It is necessary to derive the mode equation of Therefore, a method for deriving a mode equation of non-viscous sloshing by applying the Galerkin method (mode expansion method) to the variational principle in the case of a perfect fluid (non-viscous fluid) is described in Sections 1.1 to 1.3. I will explain it. Here, the necessity to use the variation principle arises from determining the damping ratio to be introduced into the mode equation of inviscid sloshing based on the variation principle.

1.1 Introduction of spherical coordinates Consider the case of a spherical tank 1 as a general axisymmetric tank arranged in a low-gravity field as shown in FIG. In FIG. 1, a meniscus is a liquid level at the time of static equilibrium of the liquid 2 stored in the tank 1, and when there is no surface tension, it is a plane perpendicular to the z-axis, but has a surface tension. In this case, an axisymmetric curved surface as shown in FIG.

を導入し、静的液面（メニスカス）Ｍ、振動液面Ｆ、タンク壁面Ｗの半径座標を、角座標の関数として次のように表す。

In order to make it possible to apply the analytical method to general axisymmetric tanks other than the cylindrical tank, the following spherical coordinates are set with the origin O as the apex of the cone that touches the tank wall surface at the contact intersection line between the meniscus and the tank 1 wall surface.

And the radial coordinates of the static liquid surface (meniscus) M, the vibrating liquid surface F, and the tank wall surface W are expressed as a function of angular coordinates as follows.

  In the above description, the origin O of the spherical coordinate is on the upper side of the tank 1 as shown in FIG. 1 when the differential with respect to z of the r coordinate of the tank 1 wall surface is negative in the contact line with the tank 1 wall surface of the meniscus. It is. When this differentiation is positive, the origin O of the spherical coordinates is below the tank.

を導入することにより解析的方法を開発している。 In general axisymmetric tanks other than cylindrical tanks, it is difficult to apply the analytical method even in the case of non-viscous sloshing. However, in Non-Patent Document 1 described above, spherical coordinates as shown in FIG.

Analytical methods are being developed by introducing.

1.2 Variation principle If the surface tension of the liquid 2 in the tank 1 is not taken into account, the Lagrangian density of the liquid 2 (the Lagrangian per unit volume) is equal to the hydraulic pressure, so the variation principle is .

[] In the formula (1.4) is Lagrangian. The variational principle when considering the surface tension is Lagrangian = (kinetic energy)-(potential energy), and therefore it is necessary to subtract the potential energy due to atmospheric pressure and interfacial tension, so that .

となる。 The hydraulic p _l, expressed as a velocity potential of the liquid 2,

It becomes.

By substituting the above equation (1.6) into the equation (1.5) and taking variations for the velocity potential, liquid level displacement, and arbitrary time function, the following equation is obtained.

Here, the definition is as follows.

In the above, from the arbitrary independence of the variations δφ, δζ, and δG with respect to velocity potential, liquid level displacement, and arbitrary time function, Equation (1.7) gives the following governing equation E ₁ = 0, etc. The physical meaning of the governing equation system is as follows.
E ₁ = 0: Continuous condition in the liquid region V (Laplace equation).
E ₂ = 0: Kinematic boundary condition on the wall surface 1 of the tank 1 (conditions in which the normal direction component of the liquid surface of the flow velocity is 0 in order to make the tank 1 wall surface a rigid body).
E ₃ = 0: Kinematic boundary condition at the vibrating liquid level F (a condition in which the components in the normal direction of the liquid level of the flow velocity and the liquid surface vibration velocity are equal).
E ₄ = 0: Mechanical boundary condition on the vibrating liquid surface F (balance condition between the surface tension and the difference between the hydraulic pressure and the atmospheric pressure).
E ₅ = 0: Balancing condition between the interfacial tensions on the contact line C (intersection line with the tank surface of the moving liquid surface, not the liquid surface at the time of static equilibrium). Since the contact angle θ _C ′ between the moving liquid surface and the tank wall surface is determined by the interfacial tension, it is called a contact angle condition.
E ₆ = 0: a constant volume condition of liquid 2 based on the incompressibility of liquid 2 (this condition can be derived from other kinematic conditions).

Next, the above formula (1.7) is expressed by the spherical coordinates introduced in the above section 1.1. That is, by using the differential geometry theorem, normal vectors N _F , N _W ; curved surface elements dF, dW; line elements dC and cos θ ′ _C are expressed as R in Expressions (1.2) and (1.3). _F, represented by _{R W.} Further, based on the linear micro-amplitude theory, the boundary condition on the liquid surface is linearly approximated, and the static balance condition used for determining the shape of the meniscus is taken into consideration, whereby the equation (1.7) of the variation principle is used. ) Is expressed in spherical coordinates as follows.

For convenience of calculation, the following quantities are used for non-dimensionalization.
Representative length b ^* (half the height of tank 1, see Fig. 1)
Representative frequency ω _ch ^* (defined by equation (1.12) described later)
Liquid density ρ _f ^*

That is, the relationship between a dimensional quantity and a dimensionless quantity is defined as follows.

  In addition, as in the above formula (1.10) and formula (1.11), the dimensional quantity is distinguished from the dimensionless quantity by adding *.

Bo defined by the above formula (1.11) is a dimensionless number representing the relative strength of gravity g ^* and surface tension σ ^* , and is called the bond number. In accordance with the number of bonds Bo, the representative frequency ω _ch ^* is defined as in the following formula (1.12), and a constant as in the formula (1.13) is introduced.

The variational principle (equation (1.9)) of sloshing in a general axisymmetric tank with dimensionless display is as follows.

1.3 Non-viscous Sloshing Mode Equation By using the spherical coordinates as introduced in section 1.1 above, the velocity potential and liquid level displacement satisfying the Laplace equation are also analyzed for the general axisymmetric tank as follows: (See Non-Patent Document 1).

In the above, if the eigenvalue λ _k is determined by the following boundary condition, a characteristic function and a characteristic index are determined.

  In the contact line between the liquid level and the tank wall surface, the θ direction is the normal direction of the tank wall surface, so the above equation (1.18) is a boundary condition where the tank wall normal direction component of the flow velocity is zero.

Therefore, the following points are features of the low-gravity sloshing attenuation ratio prediction method of the present invention.
First, since the maximum value of θ is less than 90 degrees, the spherical coordinate used in the present invention defined by the cone in FIG. 1 is officially expressed as a Legendre polynomial as a solution of the characteristic function of the equation (1.17). Therefore, it is necessary to obtain a new infinite series solution (Gaussian hypergeometric series).
Secondly, the infinite series has a convergence ability up to θ of less than 180 degrees, but as described above, θ is less than 90 degrees as shown in FIG. 1, so that convergence is fast and facilitates high-speed calculation by an analytical method.

とおいた式を上記式（１．１４）に代入し、未定定数について変分をとることによって、次のような式を導く。

In order to derive a mode equation using the equation (1.15) to the equation (1.14), first, free vibration analysis is performed. In the above formula (1.15)

By substituting the above equation into the above equation (1.14) and taking variation for the undetermined constant, the following equation is derived.

In order to have a solution other than a meaningless solution in which all undetermined constants are 0, the coefficient determinant of the above equation (1.20) needs to be 0. From this condition, the natural frequency ω and the natural mode Find the ratio between the undetermined constants. By substituting the equation obtained by substituting the result into the equation (1.15) into the equation (1.14) and taking the variation with respect to the mode coordinates, the following mode equation is derived.

1.4 Damping Analysis The non-viscous sloshing mode equation (1.21) derived in the above-mentioned sections 1.1 to 1.3 is a first-order derivative term with respect to the time of the mode coordinate q (t) corresponding to the velocity term ( Since no q-dot term is included, no attenuation occurs.

  Therefore, next, the non-viscous sloshing mode equation (1.21) is expanded to include the viscous boundary layer, thereby reducing the low gravity sloshing of the liquid 2 stored in the tank 1 shown in FIG. The attenuation due to the viscosity of the liquid 2 is analyzed.

The velocity component in the tangential direction of the tank 1 wall surface just outside the boundary layer formed along the wall surface of the tank 1 (liquid side) is calculated from the velocity potential of the equation (1.15) by the following equation, and the mode coordinates q (t ) With time differentiation (q dots).

  At this time, in order to obtain a virtual work display of the viscosity term of the Navier-Stokes equation, it is necessary to represent the flow velocity component in a curved coordinate system in which the Navier-Stokes equation can be described.

では、Ｒ方向と境界層のなす角が０度から９０度まで大きく変わり得る。そこで、図１に示すように、タンク中心に原点を有する新たな球座標

を導入する。 Note that the spherical coordinates in FIG.

Then, the angle formed by the R direction and the boundary layer can vary greatly from 0 degrees to 90 degrees. Therefore, as shown in FIG. 1, a new spherical coordinate having an origin at the center of the tank.

Is introduced.

Coordinates (dimensionalless) taken inward in the normal direction of the tank 1 wall surface are defined as follows with the tank wall surface as the origin.

は、上記式（１．２２）を次式に代入することによって求められる（タンク１壁面に垂直な流速成分が０であることを考慮している）。

The dependence of the boundary layer flow velocity on ξ can be approximated by the flow velocity of an oscillating flow along a flat plate having a frequency equal to the sloshing frequency ω because the boundary layer is very thin. Therefore, the flow velocity in the boundary layer

Is obtained by substituting the above equation (1.22) into the following equation (considering that the flow velocity component perpendicular to the wall surface of the tank 1 is zero).

The following parameters are introduced as a measure of the dimensionless boundary layer thickness.

The virtual work formed by the viscosity term of the Navier-Stokes equation in the boundary layer formed on the tank 1 wall is as follows.

In the above equation (1.28), the dimensionless viscosity coefficient μ has the following relationship with the viscosity coefficient μ ^* due to the non-dimensionalization of the above equations (1.10) and (1.11).

ここで、Ｃ_Ｓは積分によって生じる定数である。 By substituting equation (1.24) into equation (1.27) and taking the real part, the virtual work is expressed in the mode coordinates (q (t)) as follows.

Here, C _S is a constant caused by the integration.

を導く。 Adding the above formula (1.31) to the formula (1.21)

Lead.

  Since the above equation (1.32) includes a first-order differential term (q dot term) depending on the time of the mode coordinates q (t) corresponding to the velocity term, attenuation occurs.

Therefore, based on the above equation (1.32), more specifically, based on the coefficient of the first-order differential term (q-dot term) according to the time of the mode coordinate q (t), the low gravity sloshing by the following equation: The damping ratio (mode damping ratio) can be obtained.

  As described above, the low-gravity sloshing damping ratio prediction method of the present invention considers the virtual displacement of the fluid, and considering the virtual displacement in this way is based on the following idea. That is, many flow problems cannot be considered because the fluid displacement is infinite. However, in the fluid vibration problem, the displacement of the fluid can be considered, and in the present invention, it is effectively used for the evaluation of the above formula (1.27) as the virtual work of the viscous force of the liquid 2 in the tank 1. it can.

  In addition, in the process of deriving the above equation (1.33), the variation principle (refer to equation (1.5)) considering the surface tension of the liquid is used. Gravitational sloshing attenuation analysis can be performed.

  Therefore, according to the method for predicting the damping ratio of low gravity sloshing of the present invention, it is based on the viscosity of low gravity sloshing generated in the liquid 2 stored in the spherical tank 1 as a general axisymmetric tank disposed in the low gravity field. The damping ratio can be determined theoretically. For this reason, a lot of time and cost for the parameter study that is required in the method of conducting the experimental research on the low gravity sloshing of the liquid in the tank as shown in Non-Patent Document 3 is unnecessary. The analysis for obtaining the damping ratio of the low gravity sloshing can be greatly facilitated.

  Next, as an effect of the low-gravity sloshing attenuation ratio prediction method of the present invention, the validity of the mode attenuation ratio obtained based on the above equation (1.33) derived as described above is shown as a numerical example. The result verified by using will be described.

2.1 Variation of damping ratio For the liquid 2 in the spherical tank 1 shown in Fig. 1, the parameters were set as follows, and the damping ratio of low gravity sloshing was calculated based on the above equation (1.33). .
The radius a ^* = b ^* = 0.2 m of the tank 1 (that is, a spherical tank having a radius of 0.2 m as a typical example of a general axisymmetric tank), the liquid density ρ _f ^* = 1000 kg / m ^{3 of} the liquid 2, Surface tension of liquid 2 σ ^* = 0.0725 N / m, viscosity coefficient of liquid 2 μ ^* = 0.0011 Ns / m ²

At this time, the damping ratio of the low gravity sloshing when the bond number Bo, which is a dimensionless number representing the relative strength of the gravity g ^* and the surface tension σ ^* , is changed to 1000, 100, 10 and 1, respectively. Was calculated.

As described above, the bond number Bo represents the relative strength of gravity g ^* and surface tension σ ^* . Therefore, the decrease in the number of bonds Bo indicates a state where the effect (influence) of the surface tension σ ^* is relatively increased as the gravity g ^* is decreased, that is, a lower gravity field state.

  FIG. 2 shows the calculation result. The horizontal axis is the liquid filling rate (Liquid filling level: volume of liquid 2 / volume of tank 1), and the attenuation rate calculated for each number of bonds Bo (above). As for the damping ratio, the dependence on the liquid filling rate (hereinafter simply referred to as a change due to the liquid filling rate, the same applies to FIGS. 3 to 10A and 10B described later).

As can be seen from FIG. 2, when the number of bonds Bo decreases (that is, when the effect of the surface tension σ ^* becomes relatively large as the gravity g ^* decreases), the damping ratio increases.

と変形し、減衰係数（Damping coefficient）Ｃ_ｓ ^＊、質量パラメータ（Mass parameter）Ｍ_ｓ ^＊、Ｃ_ｓ ^＊／Ｍ_ｓ ^＊、及び、固有周期（Natural period）２π／ω^＊について、液体充填率（Liquid filling level）による変化を、図３、図４、図５及び図６にそれぞれ示す。なお、上記諸量に関しても、上記図２の場合と同様に、ボンド数Ｂｏを１０００、１００、１０、１にそれぞれ変化させた場合について示してある。 In order to consider this reason, an expression obtained by dimensioning the above expression (1.33)

And the liquid filling rate (Damping coefficient C _s ^* , Mass parameter M _s ^* , C _s ^* / M _s ^* , and Natural period 2π / ω ^* ) The changes due to the liquid filling level are shown in FIGS. 3, 4, 5 and 6, respectively. Note that the various quantities are also shown for the case where the number of bonds Bo is changed to 1000, 100, 10 and 1 as in the case of FIG.

From the results of FIG. 3 and FIG. 4, the following can be seen for changes in the attenuation coefficient C _s ^* and the mass parameter M _s ^* due to a decrease in the number of bonds Bo.

(1) At low liquid filling rates, the damping coefficient C _s ^* does not increase as significantly as the mass parameter M _s ^* . (2) At a high filling rate, the damping coefficient C _s ^* decreases monotonously unlike the mass parameter M _s ^* .

Therefore, C _s ^* / M _s ^* decreases with a decrease in the number of bonds Bo regardless of the filling rate as shown in FIG. However, from the results of FIGS. 5 and 6, when the number of bonds Bo is decreased as described above, the natural period 2π / ω ^* is increased more than the decrease of C _s ^* / M _s ^* (see FIG. 5) (see FIG. 5). 6) is more remarkable.

  As a result, the attenuation ratio of the above formula (2.1) proportional to these products increases as shown in FIG. 2 when the bond number Bo decreases.

This means that the increase in the damping ratio due to the decrease in the number of bonds Bo is due to the increase in the natural period 2π / ω ^* . As the natural period 2π / ω ^* increases, the attenuation ratio in FIG. 2 increases monotonously as the bond number Bo decreases, unlike the attenuation coefficient C _s ^{* in} FIG. 3.

2.2 Dependence of bond coefficient Bo on damping coefficient at high liquid filling rate As noted in the results of FIGS. 2 and 3, when the liquid filling rate is high, the damping ratio increases as the bond number Bo decreases. However, the attenuation coefficient is reduced (see FIG. 3). The reason for this can be explained as follows by finding a correlation parameter that exhibits a change similar to the attenuation coefficient.

に関する体積積分によって計算される。球形のタンク１の場合、

である。 The damping coefficient is calculated from the virtual work (see equation (1.27)) formed by the viscosity term of the Navier-Stokes equation, and this virtual work is shown in the viscous boundary layer formed along the wall surface of the tank 1. Spherical coordinates shown in 1

Calculated by the volume integral. In the case of a spherical tank 1,

It is.

である。 Therefore, first, the correlation parameter of the integral in the thickness direction of the boundary layer (specifically, a parameter that exhibits a similar change, that is, a highly correlated parameter that is easy to understand physically) is found. The boundary layer thickness order is

It is.

Since the boundary layer is very thin, among the viscosity terms of the Navier-Stokes equation, the second-order derivative in the thickness direction is dominant, and the result obtained by integrating R _{1 in the} thickness direction correlates with the following quantity.

の相関パラメータを見出す。この積分には、液面近くでの積分が大きく寄与する（タンク１壁面の接線方向の主流速度は液面近くで大きいため）。したがって、減衰係数は、液面とタンク１壁面との接触線の半径ｒ_ｃ ^＊と正の相関を有する。 next,

Find the correlation parameters. The integration near the liquid level greatly contributes to this integration (because the main flow velocity in the tangential direction of the tank 1 wall surface is large near the liquid level). Therefore, the attenuation coefficient has a positive correlation with the radius r _c ^* of the contact line between the liquid level and the tank 1 wall surface.

When showing a change due to the radius of the contact line (Contact line radius) r _c ^* liquid filling rate (Liquid filling level) in FIG. 7, in fact, from FIG. 3 and FIG. 7, when the high liquid filling rate, number of bonds As Bo decreases, both the damping coefficient and the contact line radius r _c ^* decrease. Reduction of the radius r _c ^* of the contact line, when the high liquid filling factor liquid filling rate decreases constant while the bond number Bo, when the liquid surface is curved as shown in FIG. 1 by surface tension, contact line It is understood from approaching the top of the tank 1.

Based on the above, as a correlation parameter for explaining the dependence of the damping coefficient on the bond number Bo when the liquid filling rate is high, the product of the above two correlation parameters, that is, the kinematic viscosity coefficient is constant, the radius r of the contact line The product of _c ^* and the square root of the natural frequency can be introduced. FIG. 8 shows the change of the correlation parameter r _c ^* (ω ^* / 2π) ^1/2 depending on the liquid filling rate. 3 and 8, it can be seen that the above-described correlation parameter r _c ^* (ω ^* / 2π) ^1/2 and the Bo dependency when the liquid filling rate of the damping coefficient is high are similar.

として計算され、この積分値は、メニスカス液面の面積に大きく依存する。図９に、メニスカス液面の面積（Area of meniscus）の液体充填率による変化を示す。図４と図９の比較により、メニスカス液面の面積は、質量パラメータＭ_ｓ ^＊と相関の強い指標であることが分かる。このように、減衰比に関係する質量パラメータＭ_ｓ ^＊及び減衰係数の変動が、上記所定の相関パラメータの導入によって説明される。 2.3 Correlation Parameter of Mass Parameter M _s ^* Next, the mass parameter M _s ^* shown in FIG. 4 will be considered. The kinetic energy related to the mass parameter M _s ^* is the amount of area on the meniscus liquid level M.

This integral value greatly depends on the area of the meniscus liquid surface. FIG. 9 shows the change of the meniscus liquid surface area (Area of meniscus) depending on the liquid filling rate. By comparing FIG. 4 and FIG. 9, it can be seen that the area of the meniscus liquid surface is an index having a strong correlation with the mass parameter M _s ^* . Thus, the variation of the mass parameter M _s ^* and the attenuation coefficient related to the attenuation ratio is explained by the introduction of the predetermined correlation parameter.

2.4 Comparison with Literature Experiments The damping ratio prediction results obtained by the low gravity sloshing damping ratio prediction method of the present invention are shown in Table III (p. 12, 13) of Non-Patent Literature 3 as part of the experiment (acetone as a liquid). And the data when using. In the experiment in Non-Patent Document 3, a low-gravity environment in which the number of bonds Bo is small is realized by using a small experimental model in the ground gravity field.

  Both results (data) are shown in FIGS. 10A and 10B as changes due to the liquid filling level.

The parameters used for FIGS. 10A and 10B are as shown in Table 1 below.

Here, the Galileo number Ga (dimensionless number) among the parameters shown in Table 1 is defined by the following equation.

である。したがって、非特許文献３でのボンド数Ｂｏは、表１に示した本解析でのボンド数Ｂｏ（式（１．１１）参照）の、それぞれ、３，４，３倍である。 In Non-Patent Document 3, experimental results are given for the volume of the liquid with an average liquid depth of h _av = 0.5a, a, 1.5a (a is a spherical tank radius). Also, the representative length a _ch used to define the bond number Bo is twice the r coordinate of the tank wall surface at z = h _av , that is, h _av = 0.5a, a, unlike this analysis. , 1.5a, respectively

It is. Therefore, the number of bonds Bo in Non-Patent Document 3 is 3, 4, and 3 times the number of bonds Bo (see formula (1.11)) in the main analysis shown in Table 1, respectively.

  From the results of FIGS. 10A and 10B, it can be confirmed that the analysis result of the attenuation ratio by the attenuation ratio prediction method of the low gravity sloshing of the present invention and the experimental value shown in Non-Patent Document 3 are substantially the same. .

  As described above, the physical meaning and theoretical basis of the expression derived empirically (experimental) in Non-Patent Document 3 should be theoretically explained by the attenuation ratio prediction method for low gravity sloshing of the present invention. Will be able to.

  In addition, according to this inventor, the favorable agreement with the experiment regarding the natural frequency in a nonpatent literature 1 has been confirmed.

  Therefore, according to the low-gravity sloshing damping ratio prediction method of the present invention, the sloshing attenuation ratio of the liquid 2 generated in the tank 1 disposed in the low-gravity field can be predicted to be a value significantly different from the surface gravity field. It is also clear that an effect that makes it possible to investigate the physical cause can be obtained.

  Next, FIGS. 11 and 12 show another embodiment of the present invention, using the low-gravity sloshing attenuation ratio prediction method of the present invention shown in FIGS. 1 to 10 (b) and (b). A design method for creating a low-gravity sloshing, which is a mass spring design model in the case of including damping, is shown in the following procedure.

The creation of the design model means that the design model of the tank 1 in FIG.
Condition: The frequency response of the force in the x-axis acting on the tank 1 and the moment around the y-axis to the tank vibration acceleration (f-two dots (x)) in the x-axis is the sloshing system in FIG. 1 and the design model in FIG. And are equal.
Means to determine parameters (movable mass (sloshing mass), fixed mass, mounting height of these masses, spring constant, damping coefficient) to be described later in the design model of FIG. It is.

3.1 Frequency Response of Sloshing System That is, first, as shown in FIG. 11, in the xyz coordinate space formed by the x-axis and y-axis in two directions orthogonal to each other in the horizontal plane and the vertical z-axis, When it is assumed that a spherical tank 1 as an axially symmetric tank is stored in the tank 1 with the center of the bottom of the tank 1 aligned with the origin in the xyz coordinate space, the force F _x in the x-axis direction acting on the tank 1 by the direction of sloshing along the x-axis direction of the liquid, the moment M _y around the y-axis is derived in damping ratio prediction method of low gravity sloshing of the invention described above formula Using mode coordinates q (t) obtained as a solution of (1.32), it is expressed in the following form.

  The third and fourth terms on the right-hand side of the above equations (3.1) and (3.2) are terms caused by surface tension that becomes important under low gravity.

Further, the above equation (1.32) is modified as follows.

に対する上記式（３．３）の解を、

として求め、該求められた解（式（３．６））を、上記式（３．１）及び式（３．２）に代入して、タンク１に作用する上記力Ｆ_ｘとモーメントＭ_ｙの加振加速度に対する応答を次のように求める。

Next, excitation acceleration

The solution of equation (3.3) above for

Substituting the obtained solution (formula (3.6)) into the formula (3.1) and formula (3.2), the force F _x and the moment M _y acting on the tank 1 The response to the excitation acceleration is obtained as follows.

3.2 Frequency response of the design model Next, as shown in FIG. 11, at the position of the height l ₀ on the z-axis in the tank 1, that is, at the coordinates (0, 0, l ₀ ), the mass m ₀ A fixed mass 3 is arranged to fix the fixed mass 3 to the wall surface of the tank 1 and to a position of the height l ₁ on the z axis in the tank 1, that is, at coordinates (0, 0, l ₁ ). , A movable mass (sloshing mass) 4 of mass m ₁ is disposed, and both sides of the movable mass 4 in the x-axis direction are placed on the wall surfaces of the tank 1 located on both sides of the x-axis direction, and the spring element in the x-axis direction. A design model (also referred to as a mechanical model) in which the movable mass 4 is reciprocally moved in the direction along the x-axis direction by the spring element 5 and the damper 6 is temporarily set.

The spring constant of the spring element 5 between the movable mass 4 and the wall surface of the tank 1 is k ₁ , and the damping coefficient of the damper 6 is c ₁ .

The equation of motion of the movable mass 4 in the temporary design model and the equations of the force _{Fx, mech} in the x-axis direction acting on the tank 1 and the moment My _{, mech} around the y-axis are obtained as follows.

The fourth term on the right side of the equation of moment My _{, mech in} the above equation (3.9) represents the gravitational moment due to the displacement of the movable mass 4 in the x-axis direction.

と変形し、上記式（３．１０）における上記加振加速度（式（３．５）参照）に対する解である、

を求めて、上記式（３．９）に代入することにより、上記力Ｆ_{ｘ，ｍｅｃｈ}とモーメントＭ_{ｙ，ｍｅｃｈ} の周波数応答を求めると、

となる。 Therefore, the above equation (3.8) is

And is a solution to the excitation acceleration (see Equation (3.5)) in Equation (3.10) above.

And substituting into the above equation (3.9) to obtain the frequency response of the force F _{x, mech} and moment My _{, mech} ,

It becomes.

3.3 Determination of Design Model Parameters Next, determination of parameters in the temporary design model will be described.

より、上記仮の設計モデルにおけるパラメータを下記のようにして決定する。 The above equations (3.7) and (3.13) impose the same condition for an arbitrary excitation frequency ω _f . First, the condition that the real part of the numerator and denominator of both equations is equal

Thus, the parameters in the temporary design model are determined as follows.

Specifically, the natural frequency ω _mech of the temporary design model and the mass m ₀ of the fixed mass 3 and its attachment position l ₀ to the tank 1 from the first, second, and fourth equations under the above conditions. Determine.

Next, the third formula and the fifth formula under the above conditions are modified as follows to determine the mass m ₁ of the movable mass 4 and its attachment position l ₁ to the tank 1.

In this way, parameters other than attenuation are determined for the temporary design model as follows.

となり、したがって、上記仮の設計モデルの減衰係数が次式により定まる。

Next, from the condition that the imaginary part of the denominator in the above equations (3.7) and (3.13) is equal,

Therefore, the attenuation coefficient of the temporary design model is determined by the following equation.

3.4 Correction force and correction moment Conventionally, the attenuation ratio of low-gravity sloshing has not been calculated, but if it is obtained through experiments, etc., it is conventional to adopt the determined attenuation ratio as it is as the attenuation ratio of the design model. It was a general idea.

On the other hand, the present invention is based on the difference between the imaginary parts of the numerators of the above-described equations (3.7) and (3.13) in the above-mentioned seemingly reasonable conventional idea, as shown in FIG. The sloshing system and the design model as shown in FIG. 11 suggest that the difference expressed by the following equation occurs in the force in the x-axis direction acting on the tank and the moment around the y-axis. It is characterized by creating a low-gravity sloshing design model to enable

と変形される。更に変形するため、上記式（３．１４）から得られる

を用いると、次のように変形できる。

The force Fc in the x-axis direction to correct this difference and the moment Mc about the y-axis are based on the above formula (3.16).

And transformed. For further deformation, it is obtained from the above formula (3.14).

Can be used as follows.

In the above, the terms A ₄ and C ₄ are small by numerical verification and can be omitted. Accordingly, the force Fc in the x-axis direction to be corrected and the moment Mc about the y-axis in the above equation (3.21) are the surfaces of the equations (3.1) and (3.2) that are important in low gravity. Due to the tension term (A2, C2 term and A3, C3 term) and the moment of gravity due to the horizontal displacement of the movable mass 4 in the fourth term on the right side of the equation of the moment My _{, mech in} the equation (3.9) To do.

  Therefore, in the present invention, when creating a low-gravity sloshing design model that takes into account viscous damping with low gravity, new forces and moments due to surface tension as described above are corrected.

3.5 Creation of design model that generates correction force and correction moment In the low-gravity sloshing design model of the present invention, the force in the x-axis direction by the damper and the moment about the y-axis are expressed by the above equation (3.21). a correction force F _c given, the structure of the design model (mechanical model) to be corrected moment M _c equivalent shall constitute such system is shown in Figure 12.

Specifically, as shown in FIG. 12, the position of the height l _c on the z axis in the tank 1 arranged in the xyz coordinate space, that is, the coordinates (0, 0, 1 _c ), a movable mass (sloshing mass) 4A having a mass m _c is arranged, and both sides of the movable mass 4A in the x-axis direction are placed on the wall surface of the tank 1 positioned on both sides in the x-axis direction. The movable mass 4A is attached via the spring element 5A in the direction and the damper 6A so that the movable mass 4A can be reciprocated in the direction along the x-axis direction by the spring element 5A and the damper 6A.

Furthermore, from the denominator of the above equation (3.21), the damping ratio of the system is ζ _d and the natural frequency is ω. As a result, the damping coefficient of the damper 6A interposed between the movable mass 4A and the tank 1 wall surface becomes ζ _d 2 (m _c k _c ) ^1/2 , and similarly, the movable mass 4A and the tank 1 wall surface The spring constant of the spring element 5A interposed between is set to k _c = m _c ω ² .

Under this setting, the equation of motion of the movable mass 4A is as follows.

となる。 The force F _{damp in the} x-axis direction and the moment M _damp around the y-axis by the damper 6A of the system of FIG.

It becomes.

を、上記式（３．２４）に代入すると、

となる。 This is a solution to the excitation acceleration (equation (3.5)) of the above equation (3.22).

Is substituted into the above equation (3.24),

It becomes.

Therefore, the force F _{damp in the} x-axis direction and the moment M _damp around the y-axis in the above equation (3.26) are the force F _{c in the} x-axis direction to be corrected in the above equation (3.21) and y Based on the condition that coincides with the moment M _c around the axis, the mass m _c of the movable mass 4A and its attachment position l _c in the correction model of the low gravity sloshing design model shown in FIG. 12 are determined as follows.

Thus, according to the low-gravity sloshing design model creation method of the present invention, a new discrete system is set, and the attenuation ratio obtained by experiments or the like is directly adopted as the attenuation ratio of the design model. The surface tension term (formula (3.1) and the term of A2, C2 of formula (3.2) and formula A3, C3) becomes important in low gravity (field) by modifying the design model based on the general idea of And the gravity moment term due to the horizontal displacement of the movable mass 4A (refer to the fourth term on the right side of the equation for the moment My _{, mech in} the equation (3.9)) to add the correction force and the correction moment. Therefore, a low-gravity sloshing design model that can satisfactorily give a damping ratio due to the viscosity of the low-gravity sloshing generated in the liquid 2 (see FIG. 1) in the tank 1 arranged in the low-gravity field is constructed. be able to.

Here, consider the physical meaning of the mass m _c of the movable masses 4A in the low gravity sloshing design model.

From the above equation (3.14), the following equation is derived.

By the way, the following two points are found from the description from Equation (58) to the end of Chapter 2 in Non-Patent Document 2.
(1) The mass conservation law that the right side of the above formula (3.28) is equal to the mass of the liquid is satisfied.
(2) -B ₁ is the mass of the liquid that occupies the space surrounded by the plane including the contact line with the tank wall surface of the meniscus (the plane indicated by the two-dot chain line in FIG. 1) and the tank wall surface.

Based on these points, m _c in the formula (3.27a) has a plane including the line of contact, the physical meaning of the mass of liquid filling the enclosed space in the meniscus.

  The present invention is not limited only to the above-described embodiment. In each of the above-described embodiments, the case where the present invention is applied to the spherical tank 1 has been described. However, the Laplace equation is expressed by the variable separation method using the spherical coordinates. By solving, if the velocity potential and the liquid level displacement of the liquid 2 can be expressed analytically in the same manner as in the equation (1.15), the ellipsoidal tank, the cylindrical body, and both ends of the body The invention may be applied to any type of general axisymmetric tank other than a spherical shape, such as a tank having connected hemispherical or semi-ellipsoidal end plates.

  Further, any tank other than the propellant tank of the spacecraft may be applied as long as the tank stores liquid in a low gravity field.

  Of course, various modifications can be made without departing from the scope of the present invention.

1 Tank (General Axisymmetric Tank)
2 Liquid 4A Movable mass 5A Spring element 6A Damper

Claims (2)

Non-viscous sloshing based on the variational principle including the surface tension of non-viscous fluid assuming that the general axisymmetric tank placed in a low gravity field is expressed in spherical coordinates and the liquid inside the general axisymmetric tank is non-viscous Next, in the viscosity term of the Navier-Stokes equation, the virtual work in the viscous boundary layer that can be formed on the tank wall surface is represented by mode coordinates, and the viscosity that can be formed on the tank wall surface represented by the mode coordinates. Attenuation of low-gravity sloshing characterized by obtaining a mode equation including a first-order derivative term depending on time of a mode coordinate corresponding to a velocity term in addition to the above-mentioned inviscid sloshing mode equation for virtual work in the boundary layer Ratio prediction method. Non-viscous sloshing based on the variational principle including the surface tension of non-viscous fluid assuming that the general axisymmetric tank placed in a low gravity field is expressed in spherical coordinates and the liquid inside the general axisymmetric tank is non-viscous Next, in the viscosity term of the Navier-Stokes equation, the virtual work in the viscous boundary layer that can be formed on the tank wall surface is represented by mode coordinates, and the viscosity that can be formed on the tank wall surface represented by the mode coordinates. The virtual work in the boundary layer is added to the mode equation of the non-viscous sloshing to obtain a mode equation including a first-order differential term according to the time of the mode coordinate corresponding to the velocity term, and then as a solution of the mode equation With regard to the sloshing system in which sloshing occurs in the horizontal uniaxial direction of the liquid inside the general axisymmetric tank using the obtained mode coordinates, The uniaxial force acting on the general axisymmetric tank and the moment about the axis perpendicular to the uniaxial direction in the horizontal plane are obtained, and the uniaxial force and the moment about the axis perpendicular to the sloshing system are excited. The response to acceleration is obtained, while a fixed mass is fixed at a certain height in the tank, and a movable mass arranged at another certain height is supported on the tank wall via a spring element and a damper. A temporary design model having a configuration as described above is set, and from the equation of motion of the movable mass in the design model, the uniaxial force acting on the tank of the temporary design model and the moment about the axis orthogonal thereto And the response to the same excitation acceleration of the uniaxial force and the moment about the axis orthogonal to the uniaxial direction in the design model, Response of the uniaxial force and the moment around the axis perpendicular to the sloshing system to the excitation acceleration, and the same excitation of the uniaxial force and the moment around the axis perpendicular to the temporary design model The damping coefficient required under the condition that the response to acceleration is equal for an arbitrary excitation frequency is obtained, and further, the response to the excitation acceleration of the force in the uniaxial direction and the moment about the axis orthogonal thereto in the sloshing system. And a correction force and a correction moment necessary to correct the difference between the force in the uniaxial direction in the temporary design model and the response to the same excitation acceleration of the moment about the axis orthogonal to the above, Thereafter, the movable mass is attached to the tank via the uniaxial spring element and the damper, and the Create a design model for low-gravity sloshing by creating a design model by determining the mass of the movable mass and its mounting position so that the force and moment generated by the damper are equivalent to the correction force and correction moment. Method.

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