JP2013095318A - Estimating method and calculating device of sloshing damping ratio - Google Patents

Abstract

PROBLEM TO BE SOLVED: To estimate a sloshing damping ratio by contact angle history taking energy dissipation into consideration as well.SOLUTION: An axially symmetric tank 1 is shown by spherical coordinates. A vibration equation related to a base order mode of the sloshing including the surface tension of a non-viscous fluid is derived assuming that a liquid in the axially symmetric tank 1 is non-viscous. Balance between a frictional force and the surface tension of the liquid 2 on a wall surface of the axially symmetric tank 1 is modelled taking the history into consideration. A relational expression expressing the frictional force with the change of the contact angle of the liquid 2 to the axially symmetric tank 1 is derived. Virtual work of the frictional force is evaluated, and a nonlinear damping equation is derived by adding the virtual work to the vibration equation. An expression of the damping ratio of the nonlinear damping equation to an equivalent linear equation is derived.

Description

  The present invention relates to a sloshing attenuation ratio prediction method for predicting attenuation of sloshing amplitude in a storage liquid in a tank arranged in a low-gravity field such as a spacecraft tank in consideration of a contact angle history of the storage liquid with respect to the tank, and The present invention relates to a sloshing attenuation ratio calculation apparatus.

  In general, when a tank structure is designed for a tank that stores liquid, it is indispensable to accurately predict a load due to sloshing dynamic pressure accompanying sloshing of the liquid (stored liquid) generated in the tank. For example, 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, prediction of attenuation regarding sloshing of the stored liquid is required.

  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, the interface between gas and liquid (gas-liquid interface)
Solid-liquid interface (solid-liquid interface)
Solid-gas interface (solid-gas interface)
The contact angle between the liquid surface and the tank wall surface, which is determined by the interfacial tension at, becomes important. Therefore, the attenuation is caused not only by the viscous boundary layer but also by the contact angle history.

  Here, the contact angle history is a phenomenon in which the contact angle formed between the liquid surface and the tank wall surface is larger than the value at the time of static equilibrium when the liquid level rises and becomes smaller when the liquid level falls due to friction. Even if the liquid level is the same, the contact angle varies depending on whether the liquid level is rising or descending, and the contact angle depends on the history.

  Conventionally, as a method for modeling the contact angle history, a method of imposing a spring support boundary condition around the membrane considering the liquid surface as a tension membrane is commonly used. In Non-Patent Document 1, as shown in Equation (16) and the description below, the contact angle history is expressed in a form in which the radial gradient of the liquid level displacement is proportional to the displacement. However, it is written that it is expressed by the same formula.

  However, in such a modeling method, energy dissipation due to friction cannot be taken into account, and thus the attenuation ratio based on the contact angle history cannot be predicted. In Non-Patent Document 2, p. 171, the expression (56a) and the description below that, it states that attenuation due to hysteresis (hysteresis) and energy dissipation cannot be explained.

  On the other hand, Non-Patent Document 3 discloses a non-damping vibration equation related to the sloshing of the fundamental mode of liquid that is vibrated in the axial direction in an axisymmetric container under low gravity.

Peterson, LD, Crawley, EF, and Hansman, RJ, “Nonlinear Fluid Slosh Coupled to the Dynamics of a Spacecraft”, AI Institute of America Aeronautics and Astronautics; American Aerospace Society) AIAA Journal (Spacecraft and Rocket Magazines, Proceedings) l, Vol.27, No. 9, 1989, pp. 1230- 1240 Ibrahim, RA, Pilipchuk, VN and Ikeda, T., 2001, “Recent Advances in Liquid Sloshing Dynamics”, Applied Mechanics Reviews (Review of Applied Mechanics), Vol.54, 2001, pp. 133-199 Utsumi, M., “Low-gravity Sloshing in an Axisymmetrical Container Excited in the Axial Direction”, Asme (American Society of Mechanical Engineers; American Society of Mechanical Engineers) ASME Journal of Applied Mechanics (Journals of Applied Mechanics), Vol. 67, June 2000, pp. 344-354

  In conventional sloshing modeling methods based on contact angle history, energy dissipation due to friction cannot be taken into account, and thus the damping ratio based on contact angle history cannot be predicted. For this reason, there has been a demand for a method for predicting a sloshing attenuation ratio based on contact angle history in consideration of energy dissipation and a calculation apparatus for the sloshing attenuation ratio.

  The invention of the method of predicting the sloshing damping ratio according to the present application is based on the basics of sloshing including the surface tension of a non-viscous fluid assuming that the axisymmetric tank is expressed in spherical coordinates and the liquid inside the general axisymmetric tank is assumed to be non-viscous A vibration equation related to a mode is derived, and the balance between the friction force and surface tension of the liquid on the axisymmetric tank wall surface is modeled in consideration of the history, and the friction force is changed in contact angle with the axisymmetric tank. Is derived, a virtual work of the frictional force is evaluated, a non-linear damping equation obtained by adding this virtual work to the vibration equation is derived, and an expression of the damping ratio with respect to the equivalent linear equation of the non-linear damping equation is derived. is there.

  The axisymmetric tank is spherical, and at least one of the radius of the axisymmetric tank, the density of the liquid, the contact angle, the change in the contact angle due to the friction force, and the amplitude of the liquid on the wall of the axisymmetric tank. It is preferable to calculate the value of the attenuation ratio based on two values.

  The invention of the computing device according to the present application is to express an axisymmetric tank in spherical coordinates, and assume a liquid inside the general axisymmetric tank as non-viscous, and an oscillation equation relating to a fundamental mode of sloshing including surface tension of a non-viscous fluid The relational expression between the frictional force and surface tension of the liquid on the axisymmetric tank wall surface is modeled in consideration of history, and the frictional force is expressed by a change in the contact angle of the liquid with respect to the axisymmetric tank. The virtual work of the frictional force is evaluated, a nonlinear damping equation obtained by adding this virtual work to the vibration equation is derived, and the expression of the damping ratio with respect to the equivalent linear equation of the nonlinear damping equation is derived. Storage means storing a predetermined program including a representation of a damping ratio, the size of the axisymmetric tank, the density of the liquid, the contact angle, and the change in the contact angle due to the friction force And input means for inputting at least one value of the amplitude of the liquid on the wall surface of the axisymmetric tank, and the program stored in the storage means is read and executed, and the at least one input to the input means The value of the expression of the attenuation ratio is calculated based on the value, the calculating means for determining whether or not this value exceeds the threshold value, the value of the attenuation ratio calculated by the calculating means and the result of the determination are output. Output means.

  Preferably, the axisymmetric tank has a spherical shape, and the dimension of the axisymmetric tank is a radius of the axisymmetric tank.

  According to the present invention, it is possible to predict the sloshing attenuation ratio based on the contact angle history in consideration of energy dissipation.

It is a figure which shows the coordinate system in case an axially symmetric tank is used as a calculation model. It is a diagram showing the balance of interfacial tension on the tank wall surface, showing a) during static equilibrium, b) when the liquid surface speed on the tank wall surface is upward, and c) when the liquid surface speed on the tank wall surface is downward. It is a figure which shows equivalent linear damping ratio (zeta) _eq . It is a diagram showing a coefficient C _f of the non-linear frictional damping term. It illustrates mass parameter M _s. Is a diagram illustrating a C _f / _{M s.} It is a figure which shows natural frequency (omega) / 2 (pi). It is a figure which shows the area of a meniscus. It is a figure which shows the contact line and center z coordinate of the meniscus in the low bond number. It is a figure which shows the damping ratio by a viscous boundary layer. It is a figure which shows the structure of the calculation apparatus which estimates the sloshing attenuation ratio by a contact angle log | history. It is a flowchart which shows the flow of a series of operation | movement in a calculation apparatus.

  Embodiments of a sloshing attenuation prediction method using a contact angle history and a sloshing attenuation ratio calculation apparatus according to the present invention will be described below in detail with reference to the drawings.

[1. Configuration of Embodiment]
The present embodiment is configured in the following order. First, the non-damping vibration equation related to the fundamental mode is derived by the method of Non-Patent Document 3 (Section 2.2), and the damping term due to the contact angle history is introduced into this equation by the following procedure.
(1) Accurate modeling that describes the balance between friction force and interfacial tension in consideration of the history, and derives a relational expression that expresses friction force as a change in contact angle (Section 2.3).
(2) Evaluate the virtual work of frictional force (non-conservative force) and add this virtual work to the non-damping equation (Section 2.4).
(3) Since nonlinear attenuation is caused by hysteresis nonlinearity, the attenuation ratio (equivalent attenuation ratio) of the equivalent linear system is calculated (Section 2.5).

  As a result of the example (section 3), when the gravity is reduced, the equivalent damping ratio is significantly higher than the viscous damping ratio due to the nonlinearity due to the hysteresis. For this reason, the damping ratio due to the contact angle history obtained in this specification It was found that the actual damping ratio (sum of damping due to viscosity and contact angle history) could be underestimated to 1/3 or less if is omitted.

  Furthermore, a calculation device that enables prediction by executing calculation for the sloshing attenuation ratio in consideration of the nonlinear attenuation as described above is shown as a specific example (section 4). Appendix A shows the derivation of equation (32).

[2. Method of calculation〕
[2.1 Calculation model]
Considering the calculation model of sloshing in the axially symmetric tank 1 as shown in FIG. 1, targeting an arbitrary axisymmetric tank (other than a cylindrical tank, which is frequently used as an example), which is frequently used for artificial satellites and spacecrafts. .
V: Liquid region W: Tank wall M: Liquid level during static equilibrium (meniscus)
F: Vibrating liquid level ζ: Vibration displacement of liquid level

  The meniscus is an axisymmetric surface that is strongly curved due to surface tension in a low gravity field, and a plane perpendicular to the tank symmetry axis (z axis) in a ground gravity field where the surface tension can be ignored. The motion of liquid 2 is assumed to be a vortexless flow of incompressible perfect fluid and tank 1 is assumed to be rigid. Assuming that the vibration amplitude of the liquid surface is small, linear theory is used for the liquid surface boundary condition. That is, a linear theory is used in addition to nonlinear attenuation due to contact angle history.

[2.2 Non-damped mode equations]
In the present specification, the damping term due to the frictional force causing the contact angle history is derived from the virtual work made by the frictional force. For this reason, first, it is necessary to derive the non-damped mode equation before introducing attenuation in a variational form based on the energy principle. Hamilton's principle considering the interface energy that determines the contact angle condition is given by:

  [] Represents Lagrangian. Omitting the term of potential energy due to atmospheric pressure and interfacial energy other than the term of hydraulic pressure results in the Hamiltonian principle of the usual (not low gravity) sloshing problem based on the fact that the hydraulic pressure is equal to the Lagrangian density.

を導入し、メニスカスＭ，振動液面Ｆ，タンク壁面Ｗの動径座標を角座標の関数として次のように表す。

In order to discretize Hamilton's principle [Equation (1)] by an analytical method, a spherical coordinate system as shown in FIG.

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

The origin O of the spherical coordinates is the apex of a cone that is in contact with the tank wall surface at the line of contact between the meniscus and the tank wall surface. For this reason,
ζ: The liquid level displacement set in the R direction satisfies a conforming condition (a condition in which the liquid level displacement does not penetrate or leave the wall surface on the tank wall surface). This is a device unique to this analysis method and is convenient for modeling the contact angle history of the present invention.

When the z differential of the r coordinate of the tank wall surface is positive at the meniscus contact line, the origin of the spherical coordinate is on the lower side of the tank as opposed to FIG. Therefore, the relationship between spherical coordinates and cylindrical coordinates is

The prediction of the damping ratio, is sufficient to consider the free vibration without vibration acceleration of the tank, by omitting the vibration acceleration term, representing the hydraulic pressure p _l from the pressure equation as follows.

Substituting equation (6) into equation (1) and executing the variational calculation paying attention to the fact that the liquid region V fluctuates via the liquid level displacement, the following equation is derived.

の任意独立性より下記のような支配方程式系が導かれる：

From equation (7), variation

From the arbitrary independence of, the following governing equation system is derived:

The variational principle (7) is expressed in spherical coordinates, and the mode is discretized to obtain the variational mode equation in the following form.

M _s and K _s are sloshing mass and stiffness parameters, and the natural frequency is determined by the following equation.

Using the mode coordinate q which is the solution of the mode equation, the velocity potential and the liquid level displacement are expressed by the following equations.

  The feature of Non-Patent Document 3 is that the characteristic function can be derived analytically with respect to an arbitrary axisymmetric tank and the calculation is greatly improved.

[2.3 Relationship in which frictional force is represented by change in contact angle from static contact angle]
The static balance equation for the interfacial tension is given by the following equation from FIG.

Since equation (17) means the following, it is called a static contact angle condition.

The dynamic contact angle condition at the time of sloshing is the stopping condition E ₅ = 0 [see Expression (12)] of the variation principle (1) introduced above when there is no contact angle history due to friction. By comparing these static and dynamic contact angle conditions, it can be seen that the following relationship holds when there is no contact angle history due to friction.

The balance formula between the interfacial tension and the frictional force at the time of sloshing is expressed as follows by summarizing both cases of FIGS.

In spacecraft liquid propellant sloshing, the static contact angle is small (around 5 degrees) and the change is small, so equation (18) can be approximated as follows.

The term that does not include sgn in the equation (19) disappears from the balance equation (17) at rest. Therefore, the following relational expression expressing the frictional force by the contact angle change is derived.

を接触線に沿って積分することにより、次式によって計算される：

[2.4 Derivation of attenuation term from contact angle history]
The virtual work δW _f by the frictional force is

Is integrated along the contact line by the following formula:

Substituting equation (20)

The part of the sgn function in equation (22) is obtained by substituting the mode expansion equation (16) for liquid level displacement.

での値である。この値が１となるように定数ｃ_ｋを規格化し、式（２３）を次のように変形する。

The underlined part of Equation (23) is the position of the upward liquid level displacement mode function on the tank wall surface.

The value at. The constant _ck is normalized so that this value becomes 1, and Equation (23) is modified as follows.

Substituting equation (24) into equation (22) yields:

Substituting equation (16)

となり、次式を得る。

The integral is

And the following equation is obtained.

By adding equation (26) to the left side of equation (14), the vibration equation governing the nonlinear damping behavior due to the contact angle history is derived as follows.

ここで

Equation (28) can be written as:

here

を考え、付録Ａに示す方法で、次のように等価線形減衰比を求める。

[2.5 Equivalent linear damping ratio]
For equation (29), the following equivalent linearization equation

Then, the equivalent linear damping ratio is obtained by the method shown in Appendix A as follows.

From equations (27) and (32), the equivalent linear damping ratio is proportional to:

[3. Example〕
[3.1 Evaluation result of damping ratio]
The following parameters were used for numerical calculations.

によって定めた。これは、重力と表面張力の効果の度合を表わす無次元数である。 Gravity acceleration is the number of bonds

Determined by. This is a dimensionless number representing the degree of gravity and surface tension effects.

FIG. 3 shows an equivalent linear damping ratio. The horizontal axis represents the filling rate (ratio of liquid volume to tank volume). From Figure 3, you can see the following two points:
(1) The equivalent linear damping ratio increases as the number of bonds decreases.
(2) The equivalent linear damping ratio increases monotonically with increasing filling factor, especially for small bond numbers of 1,0.1.

In order to investigate the reasons for these results (1) and (2), changes in various parameters appearing on the right side of the equation (32) are shown in FIGS.
Fig. 4: Coefficient of nonlinear friction damping term C _f
Figure 5: Mass parameter M _s
Figure 6: C _f / M _s
Figure 7: Natural frequency ω / 2π

[Consideration on 3.2 Results (1)]
First, the reason for the result (1) will be considered. 4 and 5, the following tendency is observed as the number of bonds Bo decreases.
(A) At a filling rate higher than about 0.5, C _f decreases monotonously unlike M _s .
(B) At a filling rate lower than approximately 0.5, M _s increases monotonously unlike C _f .

As a result, as shown in FIG. 6, C _f / M _s decreases with a decrease in the number of bonds (an exceptional tendency is observed only at a high filling rate of 100 or more bonds). However, as shown in FIG. 7, the natural frequency ω appearing in the denominator of the equation (32) is significantly lower than C _f / M _{s as the} number of bonds decreases. As a result, the equivalent linear damping ratio increases as the number of bonds decreases, as shown in FIG. As described above, it has been found that the result (1) in Section 3.1 is caused by a decrease in the natural frequency due to a decrease in the number of bonds.

In the next two sections, the changes in C _f and M _s shown in FIGS. 4 and 5 will be discussed.

の変化を表している。これは、式（２）、図１より、メニスカスの接触線の半径である。メニスカスの接触線の半径は、ボンド数無限大、すなわち表面張力がなくメニスカスが平面のとき、充填率０．５で最大でタンク半径に等しい。充填率一定でボンド数が低下すると、メニスカスの下に凸の曲がりが顕著になるために、メニスカスの接触線は上昇する。従って、図４のように、充填率０．５以上のときは、ボンド数が低下すると、接触線がタンクの頂上に近づき接触線の半径は減少する。また、メニスカスの接触線の半径を最大にする充填率（接触線のｚ座標がタンク半径に等しくなる充填率）は、ボンド数の低下とともに減少する。 [Discuss the change of 3.3 _{C f]}
The change in C _f shown in FIG.

Represents changes. This is the radius of the meniscus contact line from equation (2), FIG. The radius of the contact line of the meniscus is equal to the tank radius at the maximum at a filling rate of 0.5 when the bond number is infinite, that is, when there is no surface tension and the meniscus is flat. When the number of bonds decreases with a constant filling rate, a convex curve becomes conspicuous under the meniscus, and the contact line of the meniscus increases. Therefore, as shown in FIG. 4, when the filling rate is 0.5 or more, when the number of bonds decreases, the contact line approaches the top of the tank and the radius of the contact line decreases. In addition, the filling rate that maximizes the radius of the meniscus contact line (the filling rate at which the z coordinate of the contact line is equal to the tank radius) decreases as the number of bonds decreases.

を用いてメニスカス上の面積分：

で表される。この積分値はメニスカスの面積に大きく依存する。従って、質量パラメータＭ_ｓとメニスカス面積のボンド数、充填率に対する依存性は類似してくる。 [Discuss the changes in 3.4 _{M s]}
The change of the mass parameter M _s shown in FIG. 5 exhibits a change similar to the area of the meniscus shown in FIG. The reason for this is explained as follows. The mass parameter M _s represents the kinetic energy, which is the boundary condition on the tank wall

Use for the area on the meniscus:

It is represented by This integral value greatly depends on the area of the meniscus. Therefore, the dependence of the mass parameter M _s and the meniscus area on the number of bonds and the filling rate is similar.

[3.5 Consideration of result (2)]
Consider the reason for the result (2) in Section 3.1. From FIG. 7, when the number of bonds is small, 1.0 and 0.1, the dependence of the natural frequency ω on the filling rate is weak. Therefore, the equivalent linear damping ratio given by the equation (32) has the same filling factor dependency as C _f / M _s (see FIGS. 3 and 6). Therefore, the reason for the result (2) can be investigated by considering the cause of the variation of C _f / M _s .

From FIG. 6, C _f / M _s is larger at a higher filling factor than a low filling factor, and from FIG. 4, when the number of bonds is 1.0 and 0.1, the C _f is a lower filling factor than a high filling factor. It ’s big. Thus, section 3.1 of the results (2) is based on increased at a lower filling rate than _{M s} higher filling factor when the number of bonds is less 1.0, 0.1 (see Figure 5). As described in Section 3.4, the mass parameter M _s has a strong correlation with the meniscus area. Therefore, the reason for the result (2) in Section 3.1 is based on the fact that the meniscus area becomes larger at a lower filling rate than at a higher filling rate (see FIG. 8).

  The meniscus area index is the difference between the meniscus contact line and the center z-coordinate. FIG. 9 shows these z coordinates for a bond number of 1,0.1. It can be seen that the increase in meniscus area at a low filling rate when the number of bonds decreases from 1 to 0.1 is mainly due to an increase in the contact line, not a decrease in the meniscus center.

[Comparison with 3.6 viscous damping ratio]
FIG. 10 shows the viscous damping ratio ζ _vis . The viscosity coefficient is μ = 0.0011 Ns / m ² . By comparing FIG. 3 and FIG. 10, it can be confirmed that the tendency to increase with a decrease in the number of bonds is stronger in the equivalent damping ratio based on the contact angle history than in the viscous damping ratio. The reason for this can be considered as follows. The damping ratio due to viscosity is proportional to the reciprocal of the natural frequency. The equivalent linear damping ratio based on the contact angle history is not the reciprocal of the natural frequency but is proportional to the square thereof. As shown in FIG. 7, the natural frequency decreases as the number of bonds decreases. For this reason, the damping ratio due to the contact angle history is more likely to increase as the number of bonds decreases than the viscous damping ratio.

  The difference in the dependence of the damping ratio and the viscous damping ratio on the natural frequency due to the contact angle history is based on the hysteresis nonlinearity (see the last paragraph of the appendix).

は液体充填率０．６，０．９でそれぞれ２．３，３．３に達する。このように、実際のトータル減衰比は、本発明で評価した接触角履歴による減衰比を考慮しなければ、かなり過小評価される。 As the number of bonds decreases, the damping ratio due to contact angle history increases more significantly than the viscous damping ratio. As the number of bonds decreases, the actual damping ratio of damping ratio due to contact angle history = damping ratio due to contact angle history ζ _eq + viscous damping ratio ζ _vis
This means that the contribution to increases. For example, when the number of bonds is 1,

Reaches 2.3 and 3.3 at liquid filling ratios of 0.6 and 0.9, respectively. Thus, the actual total damping ratio is considerably underestimated if the damping ratio based on the contact angle history evaluated in the present invention is not taken into consideration.

[4. (Calculator)
Prediction of the sloshing attenuation ratio in consideration of the contact angle history described above can be realized by a calculation device 10 as shown in FIG. The computing device 10 includes a calculation unit 11 such as a CPU and a DSP, a storage unit 12 such as a RAM, a ROM, and a hard disk, an output unit 13 such as an LCD and a printer, and an input unit 14 such as a keyboard and a mouse. Can be used.

  A series of operations of the calculation device 11 shown in FIG. 12 is realized by the calculation unit 11 reading and executing the sloshing attenuation ratio calculation program 12a stored in the storage unit 12. The sloshing attenuation ratio calculation program 12a includes a representation of the sloshing attenuation ratio obtained by the procedure described above.

  In the first step S1, a model is set. Here, assuming a spherical axisymmetric tank, the radius of the axisymmetric tank, the density of the liquid, the contact angle, the change in the contact angle due to the friction force, and the amplitude of the liquid on the wall of the axisymmetric tank It is assumed that the model is set with at least one value. The input unit 14 receives at least one of these values as an input value. The calculation unit 11 stores the input value received by the input unit 14 in the storage unit 12.

  In step S2, the calculation unit 11 reads the input value stored in the storage unit 12, and the numerical value calculation unit 11a calculates the value of the sloshing attenuation ratio using the expression of the sloshing attenuation ratio based on the input value. Calculate numerically. The calculation unit 11 stores the obtained sloshing attenuation ratio value in the storage unit 12.

  In step S <b> 3, the calculation unit 11 reads the value of the sloshing attenuation ratio stored in the storage unit 12 and the predetermined threshold 12 b that is also stored in the storage unit 12. If the value of the sloshing attenuation ratio does not exceed the threshold value 12b in the determination unit 11b, the calculation unit 11 determines OK and ends the series of steps. On the other hand, when the value of the sloshing attenuation ratio exceeds the threshold value, it is determined as NG and the procedure is returned to the model setting in the previous step S1. The threshold value 12b can be set via the input unit 14.

  In such a series of steps, the model setting, numerical calculation, and determination loop are repeated until the sloshing attenuation ratio falls within the predetermined threshold, thereby enabling model setting that falls within the threshold. In addition, by using the expression of the sloshing attenuation ratio described above, it is possible to set a model with high accuracy in consideration of energy dissipation.

  Such calculation of the sloshing attenuation ratio includes a representation of the aforementioned sloshing attenuation ratio, such as the sloshing attenuation ratio calculation program 12a stored in the storage unit 12, and a program having model setting, numerical calculation, and determination steps. Can also be provided.

ｑの時間微分が

となる条件を課すことより、次式が導かれる。

[Appendix A Derivation of Equation (32)]
The solution of the nonlinear equation (29) is set as follows.

The time derivative of q is

The following equation is derived by imposing the following conditions.

When the equation (A2) is differentiated with respect to time, it is as follows.

Substituting equations (A1), (A2), and (A4) into equation (29) yields the following equation:

ここで、次のようにおいた。

When formulas (A3) and (A5) are put together,

Here, it was as follows.

について解くと

Formula (A6)

Solve for

Since A and Φ are time functions that vary gradually, the right side of equation (A7) can be approximated as a time average of one cycle:

が得られる。 Note that the integrand of the second equation of equation (A8) is an odd function with respect to Ψ and the integration is performed.

Is obtained.

の根の実部

より、等価線形系の減衰振動の振幅は

となり、次の関係を満たす。

Characteristic equation of equation (31)

Real part of root

Therefore, the amplitude of the damped oscillation of the equivalent linear system is

And the following relationship is satisfied.

を消去すれば式（３２）が導かれる。すなわち、振幅変動が式（Ａ９）によって与えられる非線形系の等価線形減衰比は、式（３２）によって与えられる。 From equations (A9) and (A12)

If is deleted, equation (32) is derived. That is, the equivalent linear attenuation ratio of the nonlinear system in which the amplitude variation is given by Expression (A9) is given by Expression (32).

を行うと、次式が得られる

The dependence of the damping ratio on the natural frequency ω is compared with the case of linear damping. The following changes have been made to investigate the dependence in the linear case:

To get the following equation

と比較すると、線形減衰比は次のように固有振動数の逆数に比例することが分かる。

Formula (A13)

, It can be seen that the linear damping ratio is proportional to the reciprocal of the natural frequency as follows.

  The equivalent linear damping ratio of a nonlinear system having a contact angle history is not proportional to the natural frequency but proportional to the reciprocal of its square, as shown in equation (32), and the natural frequency dependence is stronger than the linear damping ratio. I understand.

  In addition, the above-mentioned embodiment shows a specific example of the present invention and does not limit the present invention.

DESCRIPTION OF SYMBOLS 10 Calculator 11 Calculation part 12 Storage part 13 Output part 14 Input part

Claims (4)

Representing the axisymmetric tank in spherical coordinates, assuming the liquid inside the axisymmetric tank as non-viscous, deriving an oscillation equation for the fundamental mode of sloshing including the surface tension of the non-viscous fluid,
The balance between the frictional force and surface tension of the liquid on the axisymmetric tank wall surface is modeled in consideration of history, and a relational expression representing the frictional force as a change in the contact angle of the liquid with respect to the axisymmetric tank is derived,
Evaluate the virtual work of the frictional force, and derive a nonlinear damping equation that adds this virtual work to the vibration equation,
A method for predicting a sloshing attenuation ratio, comprising deriving an expression of an attenuation ratio of the nonlinear attenuation equation with respect to an equivalent linear equation.   The axisymmetric tank is spherical, and at least one of the radius of the axisymmetric tank, the density of the liquid, the contact angle, the change in the contact angle due to the friction force, and the amplitude of the liquid on the wall of the axisymmetric tank. The sloshing attenuation ratio prediction method according to claim 1, wherein the attenuation ratio value is calculated based on two values. The axisymmetric tank is expressed in spherical coordinates, and the vibration equation for the fundamental mode of sloshing including the surface tension of the non-viscous fluid is derived assuming that the liquid inside the axisymmetric tank is non-viscous. The balance between the frictional force and the surface tension of the liquid is modeled in consideration of the history, and a relational expression representing the frictional force by a change in the contact angle of the liquid with respect to the axisymmetric tank is derived, and the virtual work of the frictional force is calculated. A predetermined program including a representation of the damping ratio obtained by evaluating and deriving a nonlinear damping equation obtained by adding this virtual work to the vibration equation and deriving a representation of the damping ratio for the equivalent linear equation of the nonlinear damping equation. Stored storage means;
Input means for inputting at least one value of the dimension of the axisymmetric tank, the density of the liquid, the contact angle, the change in the contact angle due to the frictional force, and the amplitude of the liquid on the wall of the axisymmetric tank;
The program stored in the storage means is read and executed, the value of the expression of the attenuation ratio is calculated based on the at least one value input to the input means, and whether or not this value exceeds the threshold Computing means for determining
An apparatus for calculating a sloshing attenuation ratio, comprising: an output means for outputting the value of the attenuation ratio calculated by the arithmetic means and the result of the determination.   4. The sloshing attenuation ratio calculation apparatus according to claim 3, wherein the axisymmetric tank is spherical, and the dimension of the axisymmetric tank is a radius of the axisymmetric tank.

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