JP2004338445A - Mechanical model forming method of sloshing in tear drop type tank in consideration of coriolis acceleration - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To form a designing mass spring model in consideration of a coriolis acceleration against a nonaxisymmetry problem such as sloshing of liquid in a tear-drop type tank. <P>SOLUTION: It is made possible to count in coriolis acceleration by making a third axis [z'] of a spherical coordinate cross with a liquid level M roughly at right angles near the center of the liquid level M and displaying liquid movement not limited to a potential current by making display of solution be an orthogonal function in any coordinate directions, a designing mass spring model to approximately display frequency characteristics that the tank 1 receives in each of the axes [x'], [y'] direction when the tank 1 is excited in the first and second axis [x'], [y'] directions roughly orthogonal in parallel with the liquid level M of a sloshing system is separately formed, characteristic frequency of the sloshing separated by coriolis acceleration is evaluated and a spring constant of each of the designing mass spring models is corrected so as to approximately display this evaluated separated characteristic frequency. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

に対する、タンクに作用する〈ｘ〉方向の力と〈ｙ〉軸回りのモーメントの周波数特性
を求める（ステップＩＩ〜Ｖ）。
（４）このスロッシングに対して図２のメカニカルモデル５を考え、以下の条件：〈ｘ〉方向のタンク加振加速度
【外２】

を受けた時の、タンクに作用する〈ｘ〉方向の力と〈ｙ〉軸回りのモーメントの周波数特性が、（３）で求めたスロッシング系のものと一致する
を満たすようにメカニカルモデル５の以下のパラメータを定める（ステップＶＩ）。
ｍ_１、ｍ_０：スロッシュマス、固定マス
ｌ_１、ｌ_０：スロッシュマス、固定マスの取付け位置
ｋ_１：スロッシュマス取付けばね６のばね定数
（５）方向のタンク加振加速度
【外３】

に対しても上の（３），（４）同様にメカニカルモデル５を作成する。
（６）コリオリ加速度により分離したスロッシングの固有振動数を評価する。このために非ポテンシャル流にも適用可能な変分原理、解の表示を導入する。
（７）（４），（５）のメカニカルモデル５を、（６）で評価したスロッシング周波数特性を近似的に表わせるものに修正する。具体的には、メカニカルモデル５にコリオリ加速度が作用したときに生じる２加振方向の連成項に固有振動数の分離度合いを制御する定数を導入し、２加振方向のばね６のばね定数を（６）で評価したスロッシングの分離固有振動数に合わせて修正する（ステップＶＩＩ、ＶＩＩＩ）。
【００３３】
以上が本発明の実施の形態において推薬液のスロッシングの設計マスばねモデル５（メカニカルモデル）の作成を行う際の概要であり、これらは［Ｉ］〜［ＶＩＩＩ］のステップに分けられる。
【００３４】
即ち、
［Ｉ］球座標系の導入
［ＩＩ］液体の運動方程式
［ＩＩＩ］静的液面形状
［ＩＶ］重力、遠心力下での変分原理
［Ｖ］重力、遠心力下でのスロッシング解析
［ＶＩ］重力、遠心力下でのメカニカルモデル作成
［ＶＩＩ］コリオリ加速度による分離固有振動数の算定
［ＶＩＩＩ］修正メカニカルモデル
【００３５】
次に、上記［Ｉ］〜［ＶＩＩＩ］のステップについて詳述する。
【００３６】
スロッシング解析は以下の仮定下で行う。（ｉ）液体は非圧縮完全流体、タンクは剛体とする。（ｉｉ）回転座標系からみた流速成分、液面の静的平衡位置からの振動変位は微小で、線形理論が適用できる。
【００３７】
［Ｉ］球座標系の導入
液滴形タンクに関する液体運動解析は、液体の静的形状が非軸対称であるため解析的取り扱いが困難であり、ＣＦＤに頼るのが通例であるが、以下のように球座標を設定すれば液体運動の特性関数系を解析的に決定して直交関数展開法を適用することができ、これによって、ＣＦＤに比べ計算時間、コストが著しく低減できる利点が得られる。まず、図３のように
（［ｘ’］、［ｙ’］、［ｚ’］）；［ｚ’］軸が液面の中央付近と大体直角に交わる直交座標系
を設定し、［ｚ’］軸回りに周方向座標φをとり、φの０度から３６０度までの区間をＮ個の区間に分割する。尚本文中のφは、下記各式中においては
【外４】

と記載するが、同一のものを表わす。その各区間
φ_ｊ−０．５Δφ≦φ≦φ_ｊ＋０．５Δφ

について、液面Ｍとタンク壁面４との交線でタンク壁面４に接線を引き（平面φ＝φ_ｊ内で）、この接線と［ｚ’］軸との交点を原点として球座標ＯＲθφを設定する（周方向分割区間を表わす添え字ｊは数式の煩雑さを避けるため省略する）。このような球座標設定の例を２つのｊの値について図３に示す。このような球座標を用いて、静的平衡時の液面Ｍ、振動している液面Ｆ、タンク壁面Ｗを
Ｍ：Ｒ＝Ｒ_Ｍ（θ）， Ｆ：Ｒ＝Ｒ_Ｆ（θ，φ，ｔ）＝Ｒ_Ｍ（θ）＋ζ（θ，φ，ｔ），Ｗ：Ｒ＝Ｒ_Ｗ（θ） …（２ａ，ｂ，ｃ）
と表わす。ここで
Ｒ_Ｍ（θ）：静的平衡時の液面のＲ座標
Ｒ_Ｗ（θ）：タンク壁面のＲ座標
ζ（θ，φ，ｔ）：液面の振動変位
【００３８】
式（２）のような表わし方には以下の利点がある。すなわち、液面変位がタンク壁面でタンク壁面に沿う（接する）べきであるという運動学的適合性を、タンク壁面が湾曲していても変位ベクトルの１方向成分（Ｒ成分）のみの設定で満足でき、幾何学的取り扱いが便利である。
【００３９】
ひとつ注意すべきことは、図３で原点Ｏがあるφでタンクの上側から下側に飛び移ることがあり、このようなφで＋Ｒ方向に定義された液面変位ζが不連続に変化することである。これに対しては、原点がタンクの上側、下側のどちら側にあるかに応じて−１，１をとるパラメータεを導入してζを符号反転させ、−εζ（液体領域から外側への液面変位）を連続関数とみて特性関数で展開する配慮を施す（式（１８））。
【００４０】
２つの座標系（図３参照）
（［ｘ’］，［ｙ’］，［ｚ’］）：球座標参照座標系
（［ｘ］，［ｙ］，［ｚ］）： タンク参照座標系
の関係は、原点移動とオイラー（Ｅｕｌｅｒ）角による回転（図４）で一般的に与えておく：
【数１】

ここで、回転に対応する行列成分の一例は
Ｅ_１１＝ｃｏｓφ_Ｅｃｏｓθ_ＥｃｏｓΨ_Ｅ−ｓｉｎφ_ＥｓｉｎΨ_Ｅ，
Ｅ_１２＝ｃｏｓφ_Ｅｃｏｓθ_ＥｓｉｎΨ_Ｅ−ｓｉｎφ_ＥｓｉｎΨ_Ｅ，Ｅ_１３＝ｃｏｓφ_Ｅｓｉｎθ_Ｅ …（３ｂ）
（φ_Ｅ，θ_Ｅ，Ψ_Ｅ）：オイラー角
【００４１】
［ＩＩ］液体の運動方程式
液体の運動方程式は次式で与えられる。
（δｖ_ｘ）／（δｔ）−２Ωｖ_ｙ＋１／ρ（δＰ）／（δｘ）＝０，（δｖ_ｙ）／（δｔ）＋２Ωｖ_ｘ＋１／ρ（δＰ）／（δｙ）＝０，（δｖ_ｚ）／（δｔ）＋１／ρ（δＰ）／（δｚ）＝０ …（４ａ，ｂ，ｃ）
ここで
Ｐ＝ｐ−ρｇｚ−（１／２）ρΩ^２（ｘ^２＋ｙ^２） …（５）
ｖ_ｘ，ｖ_ｙ，ｖ_ｚ：流速成分
ｐ：圧力
ρ：液体の密度
尚本文中のδは、下記各式中においては
【外５】

と記載するが、同一のものを表わす。
【００４２】
運動方程式（４）をステップ［Ｉ］で導入した球座標に変換し、加振加速度による慣性力を考慮して次式を得る。
【００４３】
【数２】

ここで

【００４４】
［ＩＩＩ］静的液面形状
スロッシング解析に先立ち、遠心力と重力との静的平衡条件によって決まる静的液面形状を求めておく必要がある。運動方程式（６）で、流速と加振加速度を０におき、圧力ｐを静圧ｐ_ｓｔとみなしてつりあい方程式に帰着させる。この式を積分してｐ＝Ｃ_１（Ｃ_１は積分定数）すなわち
ｐ_ｓｔ＝ρ［ｇｚ＋（１／２）Ω^２（ｘ^２＋ｙ^２）］＋Ｃ_１ …（８）
を得る。ｐ_ｓｔが未知形状の液面ｚ＝ｚ_ｓｔ（ｘ，ｙ）において０になる条件より、液面形状を表わす関数形ｚ_ｓｔ（ｘ，ｙ）が決定される：
ｚ＝ｚ_ｓｔ（ｘ，ｙ）＝−（Ω^２／２ｇ）（ｘ^２＋ｙ^２）−（Ｃ_１／ρｇ） …（９）
定数Ｃ_１はタンクの液体充填率より定まる。式（９）より、静的液面は回転放物面である。
【００４５】
［ＩＶ］重力、遠心力下での変分原理
式（６）で圧力ｐを以下のように表わす：
ｐ＝ｐ_ｓｔ＋ｐ_ｓｌｏ＋ｐ_ｉｍｐ …（１０）
ここで
ｐ_ｓｔ： ステップ［ＩＩＩ］で決定した静圧。
ｐ_ｓｌｏ： 液面の振動によって発生するスロッシュ圧（未知）。
ｐ_ｉｍｐ： 衝撃圧力。タンクの強制加速度によって液体が一体となって剛体移動する際の慣性によって生じ、次式によって与えられる。
【数３】

ここで液面Ｍ上での平均圧力０の条件より
【数４】

【００４６】
式（１０）を運動方程式（６）に代入して静的な関係（８）を考慮し、コリオリ加速度を省略すると
（δ／δｔ）｛ｖ_Ｒ，ｖ_θ，ｖ_φ｝^Ｔ＝−ｇｒａｄ（ｐ_ｓｌｏ／ρ） …（１３）
なる関係が導かれるので、スロッシュ圧と
（δ／δｔ）Φ＝−ｐ_ｓｌｏ／ρ …（１４）
の関係にある速度ポテンシャルΦが存在する。そこで、速度ポテンシャルで表わされた変分原理［文献（１）の式（１９）、文献（２）の式（１８）］を適用し、さらに文献（３）ｐ．６００にならい速度ポテンシャルΦの代わりにスロッシング圧力ｐ_ｓｌｏと流体変位成分
【数５】

で表わして時間に関する部分積分を行う。そして、変分原理中に現れる法線ベクトルなどを球座標で表示し、液面境界条件を線形近似して次式を得る。
【００４７】
【数６】

（ｆ_４（θ，φ），ｆ_５（θ，φ）：タンクの位置によって決まる関数、表記省略）
【００４８】
［Ｖ］重力、遠心力下でのスロッシング解析
この変分原理をガレルキン法によりモード方程式に変換する。解を解析的に表示するためにステップ［Ｉ］で導入した球座標が効力を発揮する。圧力の解をラプラス方程式の解としてもとめ、次にこの結果を式（１３）に用い時間積分して流体変位成分の解を導く：
【数７】

ここで、右辺に現れる特性関数系は下記の通りである。
【００４９】
【数８】

ここで
Ｌ_ｌ：規格化定数
λ_ｍｋとΘ_ｍｋ：次の境界値問題の解として定まるｋ番目の固有値、固有関数
Θ’’＋ｃｏｔθΘ’＋［λ−（ｍ^２／ｓｉｎ^２θ）］Θ＝０ …（２２）
Θ’＝０ ａｔ θ＝θ_ｍａｘ …（２３）
α_ｍｋ１：方程式α（α＋１）＝λ_ｍｋの２根、すなわち
α_ｍｋ１＝１／２［−１−（１＋４λ_ｍｋ）^１／２］，α_ｍｋ２＝１／２［−１＋（１＋４λ_ｍｋ）^１／２］
【００５０】
式（１７），（１８）を変分原理（１６）の強制加速度項を省略した式に代入し、未定定数Ａ_ｍｋｌｑ，Ｃ_ｍｋｑの変分に関する停留条件を課して、これらの未定定数に関する同次１次方程式系を導く。これは固有値問題の形
（−ω^２Ｍ＋Ｋ）Ｘ＝０ …（２４）
（Ｘ：Ａ_ｍｋｌｑ，Ｃ_ｍｋｑを並べた列ベクトル）
になり、さらにＣ_ｍｋｑのみに関する固有値問題に縮小できて、これを解くことにより固有振動数とモード関数を決定する。
【００５１】
メカニカルモデル５（図２参照）を
２つの加振方向：〈ｘ〉，〈ｙ〉
について求めるが、各モデルともその加振方向に振動するモードのうちの最低次モードについて求める。これらのメカニカルモデル５は、独立かつ同様に求められるので、図２紙面内の加振方向についてのみ説明する：解をモード関数で次のように表わす。
【００５２】
【数９】

【００５３】
式（２５−２７）を変分原理（１６）に代入してモード座標ｑ（ｔ）に関して変分をとり、次の形のモード方程式を導く：
【数１０】

タンクが液体から受ける動的な力とモーメントを、タンク壁での動圧を積分して計算すると、次のような形に表わされる：
【数１１】

（Ａ，Ｂ，Ｃ，Ｄは定数）
モード方程式（２８）の正弦波加振
【数１２】

に対する定常振動解を求め、式（２９）に代入すると力、モーメントの加振加速度に対する周波数応答が以下のように求められる。
【００５４】
【数１３】

【００５５】
［ＶＩ］重力、遠心力下でのメカニカルモデル作成
図２のメカニカルモデル５が同様な加振（３０）を受けたときのスロッシュマスの運動方程式、タンクに作用する力、モーメントを計算し、以下の式を得る。
【００５６】
【数１４】

ここで

【００５７】
式（３２），（３３）より、メカニカルモデルの力、モーメントの正弦波加振（３０）に対する応答を計算すると
【数１５】

ここで
ω_{〈ｘ〉，ｍｅｃｈ}＝［（ｋ_１〈ｘ〉−ｍ_１〈ｘ〉Ｐ_０〈ｘ〉）ｍ_１〈ｘ〉 ］^１／２ …（３６）
【００５８】
式（３１）の力、モーメントが式（３５）の力、モーメントに任意の加振周波数ω_ｆについて等しい条件より、メカニカルモデルの諸定数が次のように定まる。
【００５９】

【００６０】
［ＶＩＩ］ コリオリ加速度による分離固有振動数の算定
ステップ［Ｖ］で定まる固有振動数は２つの加振方向で大体等しく（図７参照）、ひとつの固有振動数ω_{０，ｓｌｏ}とみなせる。この固有振動数がコリオリ加速度の影響で２つの固有振動数ω_{１，ｓｌｏ}とω_{２，ｓｌｏ}に分離する。本ステップ［ＶＩＩ］ではメカニカルモデル修正のための第一歩として、これらの値を予測する。コリオリ加速度のある場合にはラプラス方程式の理論が使えないので、より一般的な変分原理と解の表示が必要になる。まず、変分原理として、運動方程式（６）を変分表示して式（１６）に付加した次式を用いる。
【００６１】
【数１６】

ここで
【数１７】

【００６２】
次に、解の表示について考える。ラプラス方程式が使えないので、Ｒに依存する部分にも一般的な直交関数を使いたい。しかしＲの上下限Ｒ_Ｗ、Ｒ_Ｍが一定でなくθの関数として変化するのが問題である。このような場合の従来からの頻用手法は、θの定義域を多くの区間に分割し、その各区間ごとに解を、局所的に一定な不連続固有値と、独立な一般化座標で表わし、これらの一般化座標を隣接区間との境界での連続条件から決める方法であった。このような方法とは異なり、本評価法では次式
【数１８】

のように解を表わし、固有値に相当するｃｏｓの中の
π（ｌ−１）／（Ｒ_ｗ−Ｒ_Ｍ） …（４３）
を連続関数として扱いその微分を考慮することによって隣接区間との境界での連続性を満たさせ、θの定義域全体に渡って共通の一般化座標
ａ_ｍｋｌｑ，ｂ_ｍｋｌｑ，ｃ_ｍｋｌｑ，ｄ_ｍｋｌｑ，ｅ_ｍｋｑ
を使うことによって自由度数、計算時間、コストの低減を計る。式（４１），（４２）を式（４０）に代入し、一般化座標に関する変分計算を実行すると
（−ω^２Ｍ_ｓ＋ｉωＣ_ｓ＋Ｋ_ｓ）Ｘ_ｓ＝Ｆ_ｓ …（４４）
の形の行列方程式が導かれる。ここで
Ｍ_ｓ，Ｋ_ｓ：質量および剛性行列
Ｃ_ｓ：コリオリ加速度項に由来する逆対称行列
Ｘ_ｓ：一般化座標を並べた列ベクトルである
【００６３】
これを解いて共振ピ−クの現れる周波数を捜すと、スロッシングの基本次固有振動数がコリオリ加速度によって２つに分離した値ω_{１，ｓｌｏ}とω_{２，ｓｌｏ}が分かる。
【００６４】
［ＶＩＩＩ］ 修正メカニカルモデル
ひとつのマスｍ_１が加振２方向にそれぞれ別のばね定数のばね６でタンクにつながれ、タンクがｚ軸回りにスピンを受けた場合を考え、このマスの加振２方向の運動方程式を導くと、
【数１９】

ここで
ｋ_１〈ｘ〉， ｋ_１〈ｙ〉：加振２方向のばね定数
ｕ，ｖ：加振２方向の動的変位
【００６５】
これに基づき、ステップ［ＶＩ］のメカニカルモデル５（式（３２）ともう一方の加振方向に対して同様に導かれる式、省略）を修正したメカニカルモデル５を次のように表わす。
【００６６】
【数２０】

ここで
ｍ_１〈ｘ〉、ｍ_１〈ｙ〉：〈ｘ〉，〈ｙ〉方向の運動に関する慣性マスでステップ［ＶＩ］で求めた値とする。
ｃ： コリオリ加速度による連成項によって生ずる固有振動数の分離の度合いをコントロールする定数；
ｋ’_１〈ｘ〉、ｋ’_１〈ｙ〉：分離した固有振動数の値を調整するための修正ばね定数
【００６７】
そして力、モーメントは次式によって与えられる：
【数２１】

【００６８】
式（４６）の振動数方程式は

【００６９】
式（４８）のω^２に関する２根の平均と差が、ステップ［ＶＩＩ］で求めた分離固有振動数の２乗の平均と差にそれぞれ等しい条件を課すと
（−Ｂ）／（２Ａ）＝（ω_{１，ｓｌｏ} ^２＋ω_{２，ｓｌｏ} ^２）／２，（（Ｂ^２−４ＡＣ）^１／２）／Ａ＝ω_{２，ｓｌｏ} ^２−ω_{１，ｓｌｏ} ^２ …（５０ａ，ｂ）
式（５０ａ），（５０ｂ）よりＢを消去して
Ｃ／Ａ＝ω_{１，ｓｌｏ} ^２ω_{２，ｓｌｏ} ^２ …（５１）
を導き、式（４９）を代入すると、次のようになる。
【００７０】
ω_{１，ｓｌｏ} ^２ω_{２，ｓｌｏ} ^２＝（ｋ’_１〈ｘ〉−ｍ_１〈ｘ〉Ω^２ｃｏｓ^２〈γ〉）／ｍ_１〈ｘ〉・（ｋ’_１〈ｙ〉−ｍ_１〈ｙ〉Ω^２）／ｍ_１〈ｙ〉 …（５２）
【００７１】
式（５２）の右辺の第１、２因数が積ω_{１，ｓｌｏ}ω_{２，ｓｌｏ}に等しくなるようにｋ’_１〈ｘ〉、ｋ’_１〈ｙ〉を定める。このようにして求めたメカニカルモデル５の周波数応答が、スロッシングの周波数応答の共振挙動をよく表わすことを、次節で確かめる。
【００７２】
２．適用例
まず、重力のみでスピンのない場合についてステップ［ＶＩ］のメカニカルモデルのパラメータを計算した。その結果を図５に示す。
【００７３】
図５の（ａ）は固有振動数ω_１〈ｘ〉／２π、ω_１〈ｙ〉／２π（Ｈｚ）、
（ｂ）は スロッシュマスｍ_１〈ｘ〉、ｍ_１〈ｙ〉（ｋｇ）、
（ｃ）はスロッシュマスｍ_１〈ｘ〉、ｍ_１〈ｙ〉（液体の全質量ｍ_{ｌｉｑｕｉｄ}で無次元化した値）、
（ｄ）はスロッシュマスの取付け位置ｌ_１〈ｘ〉、ｌ_１〈ｙ〉（球形部半径で無次元化した値）、
（ｅ）は固定マスの取付け位置ｌ_０〈ｘ〉、ｌ_０〈ｙ〉（球形部半径で無次元化した値）である（重力支配下； 実線、一点鎖線は〈ｘ〉、〈ｙ〉加振に対する値）。
［ａ＝０．２５ｍ，θ_Ｃ＝４０ｄｅｇ，γ＝４３ｄｅｇ，Ｌ_ｃｅｎｔ＝０．５ｍ，Ｈ_ｃｅｎｔ＝０ｍ，α＝０，Ω＝０ｒｐｍ，ｇ＝２ｍ／ｓ^２，ρ＝１００９ｋｇ／ｍ^３；
力参照座標（〈ｘ〉，〈ｙ〉，〈ｚ〉）の傾き〈γ〉＝０、原点の（ｘ’，ｙ’，ｚ’）座標（Ｌ_ｃｅｎｔ，０，Ｈ_ｃｅｎｔ＋ａ（図２参照）；
球座標参照座標（［ｘ’］，［ｙ’］，［ｚ’］）の傾き（φ_Ｅ，θ_Ｅ，Ψ_Ｅ）＝（０，−３８°，０），原点の［ｘ］，［ｙ］，［ｚ］）座標（０．１４，０，０．１７）ｍ（図３，４参照）］
【００７４】
注意すべきことは、スロッシュマスが液体充填率の増加とともに単調増加せず、途中の充填率で最大となることである。この理由は、充填率が１００％に近づくと液面の面積が小さくなって拘束が強くなり、一方、低い充填率では全液体量が小さいためスロッシュマスも小さくなるためである。以上の結果、スロッシュマスは中間より多少高い充填率で最大となる。
【００７５】
本計算法は、静的平衡時の液体形状が非軸対称な場合を対象として考案したものであるが、検証のため、円錐部を小さく（円錐部の頂角を大きく）して球形タンクに近づけた場合についてメカニカルモデルのパラメータの計算を行い、過去の文献（４）と比較して図６に示した。
【００７６】
図６の（ａ）は無次元固有振動数ω_１〈ｘ〉／（ｇ／ａ）^１／２、ω_１〈ｙ〉／（ｇ／ａ）^１／２、
（ｂ）はスロッシュマスｍ_１〈ｘ〉、ｍ_１〈ｙ〉（液体の全質量ｍ_{ｌｉｑｕｉｄ}で無次元化した値）、
（ｃ）はスロッシュマスの取付け位置ｌ_１〈ｘ〉、ｌ_１〈ｙ〉（球形部半径で無次元化した値）、
（ｄ）は固定マスの取付け位置ｌ_０〈ｘ〉、ｌ_０〈ｙ〉（球形部半径で無次元化した値）である（重力支配下、球タンクに帰着させて検証、文献（４）ｐ．１３の結果と別曲線で表わせないほど一致；実線、一点鎖線は、〈ｘ〉、〈ｙ〉加振に対する値）。
［ａ＝０．２５ｍ，θ_Ｃ＝７０ｄｅｇ，γ＝２２ｄｅｇ，Ｌ_ｃｅｎｔ＝０．５ｍ，Ｈ_ｃｅｎｔ＝０ｍ，α＝０，Ω＝０ｒｐｍ，ｇ＝２ｍ／ｓ^２，ρ＝１００９ｋｇ／ｍ^３；
力参照座標（〈ｘ〉，〈ｙ〉，〈ｚ〉）の傾き〈γ〉＝０、原点の（ｘ’，ｙ’，ｚ’）座標（Ｌ_ｃｅｎｔ，０，Ｈ_ｃｅｎｔ＋ａ）（図２参照）；
球座標参照座標（［ｘ’］，［ｙ’］，［ｚ’］）の傾き（φ_Ｅ，θ_Ｅ，Ψ_Ｅ）＝（０，−１７°，０），原点の［ｘ］，［ｙ］，［ｚ］）座標（０．０７，０，０．０１）ｍ（図３，４参照）］
【００７７】
文献（４）の結果との一致は非常によいことを確認している。
【００７８】
図７に本題のスピンのある場合のメカニカルモデルのパラメータの計算結果を示す。
【００７９】
図７の（ａ）は固有振動数ω_１〈ｘ〉／２π、ω_１〈ｙ〉／２π（Ｈｚ）、
（ｂ）は スロッシュマスｍ_１〈ｘ〉、ｍ_１〈ｙ〉（ｋｇ）、
（ｃ）はスロッシュマスｍ_１〈ｘ〉、ｍ_１〈ｙ〉（液体の全質量ｍ_{ｌｉｑｕｉｄ}で無次元化した値）、
（ｄ）はスロッシュマスの取付け位置ｌ_１〈ｘ〉、ｌ_１〈ｙ〉（球形部半径で無次元化した値）、
（ｅ）は固定マスの取付け位置ｌ_０〈ｘ〉、ｌ_０〈ｙ〉（球形部半径で無次元化した値）である（実線、一点鎖線は〈ｘ〉、〈ｙ〉加振に対する値）。
［ａ＝０．２５ｍ，θ_Ｃ＝４０ｄｅｇ，γ＝４３ｄｅｇ，Ｌ_ｃｅｎｔ＝０．５ｍ，Ｈ_ｃｅｎｔ＝０ｍ，α＝０，Ω＝２０ｒｐｍ，ｇ＝２ｍ／ｓ^２，ρ＝１００９ｋｇ／ｍ^３；
力参照座標（〈ｘ〉，〈ｙ〉，〈ｚ〉）と球座標参照座標（［ｘ’］，［ｙ’］，［ｚ’］）はタンク対象座標（［ｘ］，［ｙ］，［ｚ］）に一致（図２，３参照）］
【００８０】
スピンによって生じる液面の勾配は、液面がタンク中心線と交わる位置で
ｄｚ’／ｄｘ’
が−０．８から−１．３となる程度であり、重力が小さいためかなり大きく、コリオリ加速度による固有振動数の分離も次にみるように顕著である。
【００８１】
図８、９は、ステップ［ＶＩＩＩ］の修正メカニカルモデルによって、ステップ［ＶＩＩ］の解析で求めたスロッシングによる力とモーメントの周波数応答が近似的に表わせることを示したものである。
【００８２】
図８は
【数２２】

に対する力、モーメントの周波数応答［液体充填率４８％；実線、メカニカルモデル；点線、スロッシング系；他のパラメータは図７と同じ］を表わし、図９は
【数２３】

に対する力、モーメントの周波数応答［液体充填率４８％；実線、メカニカルモデル；点線、スロッシング系；他のパラメータは図７と同じ］を表わす。スロッシュマスが大きいため重要な中間的充填率の代表値について示した。特に分離した２つの固有振動数の間の帯域での一致がよく、２つの固有振動数の外側で多少一致が悪くなるのは、スロッシング系の計算結果では隣接するモードの寄与が含まれるためである（低周波数側にも内部振動と呼ばれるモードがある）。特に、コリオリ加速度による２加振方向の振動の連成によって、以下のことが起こる点に注意が必要である。
【００８３】
〈ｘ〉方向加振に対しても〈ｙ〉方向の力、〈ｘ〉方向のモーメントが生じる（これらはコリオリ加速度なしの場合には出ない）
〈ｙ〉方向加振に対しても〈ｘ〉方向の力、〈ｙ〉方向のモーメントが生じる（これらはコリオリ加速度なしの場合には出ない）
【００８４】
これらの連成によって生じる力、モーメント成分の位相変化が修正メカニカルモデルとスロッシング系とで一致している。
【００８５】
文献
［１］ Ｕｔｓｕｍｉ， Ｍ．， １９９８， ”Ｌｏｗ−ｇｒａｖｉｔｙ Ｐｒｏｐｅｌｌａｎｔ Ｓｌｏｓｈ Ａｎａｌｙｓｉｓ Ｕｓｉｎｇ Ｓｐｈｅｒｉｃａｌ Ｃｏｏｒｄｉｎａｔｅｓ，” Ｊｏｕｒｎａｌ ｏｆ Ｆｌｕｉｄｓ ａｎｄ Ｓｔｒｕｃｔｕｒｅｓ， １２， ｐｐ． ５７−８３．
［２］ Ｕｔｓｕｍｉ， Ｍ．， ２０００， ”Ｌｏｗ−ｇｒａｖｉｔｙ Ｓｌｏｓｈｉｎｇ ｉｎ ａｎ Ａｘｉｓｙｍｍｅｔｒｉｃａｌ Ｃｏｎｔａｉｎｅｒ Ｅｘｃｉｔｅｄ ｉｎ ｔｈｅ Ａｘｉａｌ Ｄｉｒｅｃｔｉｏｎ，” ＡＳＭＥ Ｊｏｕｒｎａｌ ｏｆ Ａｐｐｌｉｅｄ Ｍｅｃｈａｎｉｃｓ， ６７， ｐｐ． ３４４−３５４．
［３］ Ｕｔｓｕｍｉ， Ｍ．， ２０００， ”Ｄｅｖｅｌｏｐｍｅｎｔ ｏｆ Ｍｅｃｈａｎｉｃａｌ Ｍｏｄｅｌｓ ｆｏｒ Ｐｒｏｐｅｌｌａｎｔ Ｓｌｏｓｈｉｎｇ ｉｎ Ｔｅａｒｄｒｏｐ Ｔａｎｋｓ，” Ｊｏｕｒｎａｌ ｏｆ Ｓｐａｃｅｃｒａｆｔ ａｎｄ Ｒｏｃｋｅｔｓ， ３７， ｐｐ． ５９７−６０３．
［４］ ＮＡＳＡ ＳＰ−８００９， ＮＡＳＡ Ｓｐａｃｅ Ｖｅｈｉｃｌｅ Ｄｅｓｉｇｎ Ｃｒｉｔｅｒｉａ （Ｓｔｒｕｃｔｕｒｅｓ）， Ｐｒｏｐｅｌｌａｎｔ Ｓｌｏｓｈ Ｌｏａｄｓ， Ａｕｇｕｓｔ １９６８．
【００８６】
尚、本発明は上記形態例にのみ限定されるものではなく、その他本発明の要旨を逸脱しない範囲内において種々変更を加え得ること、等は勿論である。
【００８７】
【発明の効果】
本発明のコリオリ加速度を考慮した涙滴形タンク内スロッシングのメカニカルモデル作成方法よれば、以下のような優れた効果を奏し得る。
【００８８】
涙滴形タンクの加振に対するスロッシングの周波数特性の評価に際し、球座標の軸が液面と大体直角に液面の中央近くで交わるようにしてタンク円錐部のどこからでもとれるようにし、解の表示を、どの座標方向にも直交関数となる直交モード関数の重ね合わせで与えることにより、非ポテンシャル流にも適用可能にしてコリオリ加速度を含むスロッシングの周波数特性が解析的方法で評価可能となり、自由度が低減し、従って従来の有限要素法の数値的手法に比べてスロッシング系の解析、延いては設計マスばねモデル作成のための計算時間が短くなり、計算コストが安価になる。
【００８９】
従って、前記解の表示に基づき、まず、コリオリ加速度を考慮せずに、液面とほぼ平行で互いに直交する２つの加振方向について周波数特性を近似的に表わす設計マスばねモデルを別個に作成し、次に、これらの設計マスばねモデルを連成させてコリオリ加速度による固有振動数の分離が取り込めるようにし、コリオリ加速度により分離したスロッシングの固有振動数を評価して、この評価した分離固有振動数を近似的に表わせるように前記各設計マスばねモデルのばね定数を修正するようにしたので、コリオリ加速度を含む実際のスロッシング系の周波数特性に非常に近似した周波数特性を持つ設計マスばねモデルが作成できる。
【００９０】
このように、スロッシング系によるコリオリ加速度を含むスロッシングの周波数特性と近似した周波数特性を持つ設計マスばねモデルの作成が可能になることによって、衛星に対するスロッシングの影響を正確に評価して衛星の姿勢制御性を大幅に向上させることができる。
【図面の簡単な説明】
【図１】本発明を適用する涙滴形タンクとその座標系を示す概念図である。
【図２】力参照座標とメカニカルモデルの概念図である。
【図３】球座標の設定方法の概念図である。
【図４】球座標の原点と軸を決める座標変換の概念図である。
【図５】（ａ）、（ｂ）、（ｃ）、（ｄ）、（ｅ）は重力のみでスピンのない場合についてステップＶＩのメカニカルモデルのパラメータを計算した結果を示す線図である。
【図６】（ａ）、（ｂ）、（ｃ）、（ｄ）は円錐部を小さくして球形タンクに近付けた涙滴形タンクについてパラメータを計算して過去の文献と比較して示した線図である。
【図７】（ａ）、（ｂ）、（ｃ）、（ｄ）、（ｅ）は本題のスピンのある場合の計算結果を示す線図である。
【図８】（ａ）、（ｂ）、（ｃ）、（ｄ）はステップＶＩＩＩの修正メカニカルモデルによって、ステップＶＩＩの解析で求めたスロッシングによる力とモーメントの周波数応答が近似的に表せることを示した線図である。
【図９】（ａ）、（ｂ）、（ｃ）、（ｄ）はステップＶＩＩＩの修正メカニカルモデルによって、ステップＶＩＩの解析で求めたスロッシングによる力とモーメントの周波数応答が近似的に表せることを図８と異なる条件で示した線図である。
【図１０】涙滴形タンクの概要を説明するための縦断面図である。
【図１１】図１０に示す涙滴形タンクの設計マスばねモデルを図示化した概念図である。
【符号の説明】
１ 涙滴形タンク
４ タンク壁面
５ 設計マスばねモデル
６ ばね
Ｍ 液面
ｍ_１ マス
ｋ ばね定数[0001]
TECHNICAL FIELD OF THE INVENTION
The present invention relates to a method for creating a mechanical model of sloshing in a teardrop-shaped tank in consideration of Coriolis acceleration.
[0002]
[Prior art]
A satellite that navigates in low-gravity or zero-gravity space includes a satellite called a spin satellite, and the spin satellite travels while rotating.
[0003]
Thus, in the spin satellite, as a tank for storing the propellant liquid, the conical outer shell is tapered toward the direction away from the substantially hemispherical outer shell on the opening side of the substantially hemispherical outer shell. A connected so-called teardrop tank is used, an example of such a teardrop tank is shown in FIG.
[0004]
In FIG. 10, reference numeral 1 denotes a teardrop-shaped tank (hereinafter, simply referred to as a tank). The tank 1 is located at a position away from a spin axis Z which is a rotation axis of a spin satellite, and has a tapered tip of a conical outer shell. Are disposed so as to face outward with respect to the spin axis Z, and an outlet 3 for the propellant liquid 2 is provided at the tip of the tapered conical portion.
[0005]
The center axis z of the tank 1 is set so that the top of the hemispherical outer shell of the tank 1 is close to the center axis z and the outlet 3 provided in the conical outer shell is separated from the center axis z. At an inclination angle γ.
[0006]
In FIG. 10, Ω is the spin angular velocity of the tank 1 with respect to the spin axis Z of the spin satellite, a is the radius of the hemispherical outer shell of the tank 1, and L is the hemispherical outer shell of the tank 1 from the spin axis Z. The shortest distance to the point that is the starting point of the radius a of θ, θ_CIs the half apex angle of the conical outer shell of the tank 1 with respect to the central axis z of the tank 1, g is the apparent gravitational force which is the inertial force acting on the spin satellite by the propulsion acceleration of the spin satellite, and x is the cone of the tank 1. A coordinate axis passing through the vertex of the shape of the outer shell 3 on the outlet 3 side and orthogonal to the central axis z, y is a direction passing through the intersection of the central axis z and the coordinate axis x and orthogonal to the x and y axes (in FIG. (A direction orthogonal to).
[0007]
The reason why the tank 1 having the structure shown in FIG. 10 is used is to always hold the propellant liquid (liquid) 2 in the outlet 3 by using the centrifugal force generated by the spin of the spin satellite and the apparent gravity g.
[0008]
On the other hand, when a forced acceleration disturbance due to shaking or the like acts on the body of the spinning satellite during navigation, vibration (sloshing) of the propellant liquid 2 occurs in the tank 1. 1 is transmitted to the airframe, and as a result, forces and torques act on the airframe, resulting in undesirable disturbances to the attitude control system for controlling the attitude of the airframe.
[0009]
Therefore, when designing a spin satellite, it is important to design an attitude control system that takes into consideration the force and torque exerted by the sloshing of the propellant liquid 2 on the airframe. It is possible to analyze the state of sloshing of the chemical solution 2 and to create a design mass spring model (referred to as a mechanical model) having dynamic characteristics equivalent to sloshing (natural frequency and force and torque applied to the tank 1) from the analysis result. is necessary.
[0010]
However, since the liquid such as the propellant liquid 2 is a continuous body, it is difficult to handle the liquid as a control target, which is not convenient for design. Therefore, in order to facilitate the design, the propellant liquid 2 in the tank 1 is replaced with a design mass spring model as shown in FIG.₀, K, l, l₀Has been conventionally set to an appropriate value and a control system is designed based on the data.
[0011]
Here, setting to an appropriate value means that, with the replaced design mass spring model system and the original sloshing system of the propellant liquid 2, the frequency characteristics of the force applied to the tank 1 and the torque to the forced acceleration disturbance are extremely low. That is, they are set so as to match in the frequency band.
[0012]
In the parameters of FIG. 11, m is the mass of a movable object movable in the x direction, which is a direction orthogonal to the central axis z of the tank 1 in the tank 1, m₀Is the mass of the fixed object fixed in the tank 1, k is the spring constant of the movable object, l is the x-, y-, and z-axis set at the vertex of the conical outer shell of the tank 1 from the origin to the movable object. Distance of l₀Is the distance from the origin to the fixed object.
[0013]
By the way, when a design model of a mass spring system is created, sloshing analysis is necessary as a previous step. As such a sloshing analysis means, a numerical analysis method such as a finite element method (FEM) and the like have conventionally been used. A known method is known.
[0014]
In the case of a numerical analysis method, in general, in order to obtain a solution with a reasonable accuracy, a liquid to be a calculation domain is divided into a number of elements, and an equation is set using physical quantities at a number of independent points as unknown variables for each of the divided elements. Need to solve. Therefore, there is a problem that fine element division is required, the degree of freedom is enormous, the calculation time is lengthened, and the calculation cost is increased.
[0015]
On the other hand, the analytical method does not require fine element division and can reduce the calculation cost.However, the analytical method can be generally applied to the case of an axisymmetric case such as a cylindrical tank, that is, the case of the axisymmetric problem. It is.
[0016]
However, the sloshing of the propellant liquid 2 in the teardrop-shaped tank 1 as shown in FIG. 10 is a typical case where the direction of the spin axis Z and the apparent gravity g do not coincide with the central axis z which is the axis of symmetry of the tank 1 at all. It is difficult to apply the analytical method because it is a typical non-axisymmetric problem, so the calculation of the non-axisymmetric problem as described above must rely on the numerical method, and it takes a long calculation time. However, it is not possible to eliminate the adverse effect that the calculation cost increases.
[0017]
The patent applicant assumes a longitudinal section of the liquid passing through the central axis of the teardrop-shaped tank whose central axis is inclined at a predetermined angle with respect to the spin axis and a predetermined position in the circumferential direction of the liquid surface in the teardrop-shaped tank. Then, polar coordinates are taken in the vertical section of each liquid, and Laplace's equation 支配 governing the liquid motion using spherical coordinates composed of the polar coordinates and the circumferential coordinates of a portion of the liquid surface in contact with the tank wall surface is used.²The solution of Φ = 0 (Φ is a velocity potential) is analytically determined by superposition of orthogonal functions, the liquid motion is discretized by orthogonal function expansion, and the problem is reduced to a system of linear equations concerning expansion coefficients. A patent application for solving the problem and creating a design mass spring model was filed (see Patent Document 1).
[0018]
[Patent Document 1]
JP-A-11-053341
[0019]
[Problems to be solved by the invention]
In the invention of Patent Document 1, the frequency characteristic of sloshing with respect to the vibration of the teardrop-shaped tank is evaluated by taking the axis of the spherical coordinate from the apex of the tank cone, and the Coriolis acceleration is not considered. . The present invention has been considered to be satisfactory even if the Coriolis acceleration is omitted, since if the created design mass spring model is attached to the tank and rotated, the Coriolis acceleration will be received. .
[0020]
However, if the Coriolis acceleration can be quantitatively considered in the creation of the design mass spring model for the sloshing in the teardrop-shaped tank shown in Patent Document 1, if the Coriolis acceleration can be quantitatively considered, the design mass spring model more approximated to the frequency characteristics of the actual sloshing Can be created, which can contribute to improving the accuracy of the attitude controllability of the satellite.
[0021]
The present invention has been made in view of the above circumstances, and enables an analytical approach to a non-axisymmetric problem such as sloshing of a liquid such as a propellant in a teardrop-shaped tank, thereby reducing the calculation time. To provide a method of creating a mechanical model of sloshing in a teardrop-shaped tank in consideration of Coriolis acceleration, which can reduce the calculation cost and reduce the calculation cost, and can approximate the frequency characteristics of the sloshing system including the actual Coriolis acceleration with high accuracy. It was done for the purpose.
[0022]
[Means for Solving the Problems]
According to the first aspect of the present invention, a third coordinate axis substantially perpendicular to the liquid surface at substantially the center of the liquid surface of the teardrop-shaped tank whose central axis is inclined at a predetermined angle with respect to the spin axis, and the third coordinate axis An orthogonal coordinate system having first and second coordinate axes orthogonal to each other and orthogonal to each other is set, and circumferential coordinates are set around the third coordinate axis, and a plurality of circumferential coordinates between 0 degrees and 360 of the circumferential coordinates are set. The tank is divided into sections, and for each section, a tangent is drawn to the tank wall surface at the intersection of the liquid level and the tank wall surface, and spherical coordinates are set with the intersection of the tangent line and the third coordinate axis as the origin, and the teardrop is set. In the sloshing system of the shaped tank, the frequency characteristics of the force in the first coordinate axis direction and the moment about the second coordinate axis which are applied to the tank when the tank is vibrated in the first coordinate axis direction are determined in the first coordinate axis direction. Approximately express the frequency characteristics during excitation A design mass spring model is created, and the frequency characteristics of the force in the second coordinate axis direction and the moment about the first coordinate axis that the tank receives when the tank is vibrated in the second coordinate axis direction are obtained. 2. A design mass spring model that can approximately represent the frequency characteristic in the excitation in the coordinate axis direction is created, while the natural frequency of sloshing separated by Coriolis acceleration is evaluated, and the evaluated separated natural frequency is approximated. And a method of creating a mechanical model of sloshing in a teardrop-shaped tank in consideration of Coriolis acceleration, wherein a spring constant of each of the design mass spring models is corrected as shown in FIG.
[0023]
The invention according to claim 2 is characterized in that the evaluation of the natural frequency of the sloshing separated by the Coriolis acceleration is performed by introducing a variation principle applicable to a non-potential flow and a display of a solution. Item 1 relates to a method of creating a mechanical model of sloshing in a teardrop-shaped tank in consideration of Coriolis acceleration.
[0024]
According to a third aspect of the present invention, the correction of the spring constant of each of the design mass spring models is performed by changing the vibration direction of the first coordinate axis and the second coordinate axis generated when Coriolis acceleration acts on each design mass spring model. 2. The method according to claim 1, wherein a constant for controlling the degree of separation of the natural frequency is introduced into the coupled term, and a spring constant in each vibration direction is corrected in accordance with the evaluated natural frequency of the sloshing. And a method for creating a mechanical model of sloshing in a teardrop-shaped tank in consideration of Coriolis acceleration.
[0025]
According to the above means, the following operation is performed.
[0026]
When evaluating the frequency characteristics of sloshing to the vibration of a teardrop-shaped tank, the axis of the spherical coordinate intersects the liquid surface at an almost right angle near the center of the liquid surface so that it can be taken from anywhere in the tank cone, and the solution display Can be applied to non-potential flows, and the frequency characteristics of sloshing including Coriolis acceleration can be evaluated by an analytical method. Therefore, as compared with the conventional numerical method of the finite element method, the calculation time for analyzing the sloshing system, that is, for generating the design mass spring model is shortened, and the calculation cost is reduced.
[0027]
Therefore, based on the display of the solution, firstly, without considering the Coriolis acceleration, a design mass spring model that approximately represents the frequency characteristic in two vibration directions substantially parallel to the liquid surface and orthogonal to each other is separately created. Next, these design mass spring models are coupled so that the natural frequency can be separated by Coriolis acceleration, and the natural frequency of the sloshing separated by Coriolis acceleration is evaluated. Since the spring constant of each of the design mass spring models is modified so as to approximately represent the design mass spring model, a design mass spring model having a frequency characteristic very similar to the frequency characteristic of an actual sloshing system including Coriolis acceleration is obtained. Can be created.
[0028]
As described above, it is possible to create a design mass spring model having a frequency characteristic approximate to the frequency characteristic of sloshing including Coriolis acceleration by the sloshing system, thereby accurately evaluating the influence of sloshing on the satellite and controlling the attitude of the satellite. Performance can be greatly improved.
[0029]
BEST MODE FOR CARRYING OUT THE INVENTION
Hereinafter, preferred embodiments of the present invention will be described with reference to the drawings.
[0030]
1 to 4 are conceptual diagrams showing an outline of an embodiment of the present invention. FIG. 1 is a conceptual diagram showing a teardrop-shaped tank to which the present invention is applied and a coordinate system thereof. FIG. 2 is a diagram showing force reference coordinates and a mechanical model. FIG. 3 is a conceptual diagram of a method of setting spherical coordinates, and FIG. 4 is a conceptual diagram of coordinate conversion for determining the origin and axis of the spherical coordinates.
[0031]
1. Method details
[Problem Settings and Procedures]
The droplet tank 1 shown in FIG. 1 has a shape in which a conical portion and a spherical portion are joined to each other, and is installed with the generatrix of the conical portion inclined so as to be substantially parallel to the spin axis. Therefore, the propellant liquid 2 has a function of retaining the propellant liquid 2 at the outlet 3 (the apex of the conical portion), regardless of which of the centrifugal force due to the spin and the gravity due to the propulsion acceleration is dominant. In FIG.
a: Radius of spherical part
θ_C  : Half apex angle of cone
Ω: Spin angular velocity
g: gravity due to propulsion acceleration
γ: Angle (θ) between the center line of the tank and the spin axis direction_CSlightly larger corners)
L_cent, H_cent: The distance from the spin axis at the center of the spherical part of the tank, the height from the reference plane
(X, Y, Z): coordinate system fixed in space
(X, y, z): coordinate system fixed to satellite
α: Angle that defines the position of the center of the spherical part of the tank
(X ′, y ′, z ′): a coordinate system in which (x, y, z) is rotated α around the z-axis
([X], [y], [z]): tank reference coordinate system ([z] is the center line of the tank and is in the z′x ′ plane). It is obtained by rotating (x ', y', z ') by 180 degrees-[gamma] around the y' axis, and moving the origin to the vertex of the cone.
[0032]
The purpose is to create a mechanical model (design mass spring model) that can approximately represent the frequency response of the force and moment due to sloshing when a tank is vibrated in two mutually perpendicular directions that are almost parallel to the liquid surface. It is to be. Create such a mechanical model with the following steps:
(1) A force reference coordinate system (a coordinate system that determines the direction of excitation and the force and moment to be focused) is set as shown in FIG. In FIG.
(<X>, <y>, <z>): Force reference coordinate system. (X ', y', z ') is rotated 180 degrees around the y' axis, and the origin is (x ', y', z ') = (x'₀, 0, z '₀). <Γ>, x ′₀, Z '₀Is set so that the <z> axis intersects the liquid surface at a substantially right angle near the liquid surface center (see the application example later). Therefore, FIG. 2 shows a case where gravity is dominant and the liquid surface is almost horizontal.
(2) Set spherical coordinates that enable application of the analytical method (step I).
(3) In a sloshing system where Coriolis acceleration is omitted,
Tank vibration acceleration in <x> direction
[Outside 1]

Characteristics of <x> direction force and moment about <y> axis acting on tank
(Steps II to V).
(4) Considering the mechanical model 5 of FIG. 2 for this sloshing, the following conditions: tank vibration acceleration in the <x> direction
[Outside 2]

The frequency characteristics of the force in the <x> direction and the moment about the <y> axis acting on the tank when receiving the sloshing match those of the sloshing system obtained in (3).
The following parameters of the mechanical model 5 are determined so as to satisfy (Step VI).
m₁, M₀： Slosh trout, fixed trout
l₁, L₀： Slosh mass, fixed mass installation position
k₁: Spring constant of slosh mass mounting spring 6
(5) Direction tank acceleration
[Outside 3]

, A mechanical model 5 is created in the same manner as in (3) and (4) above.
(6) Evaluate the natural frequency of sloshing separated by Coriolis acceleration. For this purpose, we introduce the variational principle applicable to non-potential flows and the display of solutions.
(7) The mechanical model 5 of (4) and (5) is modified to one that can approximately represent the sloshing frequency characteristic evaluated in (6). Specifically, a constant for controlling the degree of separation of natural frequencies is introduced into a coupled term in the two vibration directions generated when Coriolis acceleration acts on the mechanical model 5, and the spring constant of the spring 6 in the two vibration directions is introduced. Is corrected in accordance with the sloshing separation natural frequency evaluated in (6) (steps VII and VIII).
[0033]
The above is the outline of creating the design mass spring model 5 (mechanical model) for sloshing of the propellant solution in the embodiment of the present invention, and these are divided into steps [I] to [VIII].
[0034]
That is,
[I] Introduction of spherical coordinate system
[II] Equation of motion of liquid
[III] Static liquid surface shape
[IV] Variational principle under gravity and centrifugal force
[V] Sloshing analysis under gravity and centrifugal force
[VI] Mechanical model creation under gravity and centrifugal force
[VII] Calculation of separated natural frequency by Coriolis acceleration
[VIII] Modified mechanical model
[0035]
Next, the steps [I] to [VIII] will be described in detail.
[0036]
Sloshing analysis is performed under the following assumptions. (I) The liquid is an incompressible perfect fluid, and the tank is a rigid body. (Ii) The flow velocity component viewed from the rotating coordinate system and the vibration displacement from the static equilibrium position of the liquid surface are minute, and linear theory can be applied.
[0037]
[I] Introduction of spherical coordinate system
In the liquid motion analysis for a droplet type tank, it is difficult to handle analytically because the static shape of the liquid is non-axisymmetric, and it is customary to rely on CFD, but if spherical coordinates are set as follows, The orthogonal function expansion method can be applied by analytically determining the characteristic function system of the liquid motion, thereby obtaining an advantage that the calculation time and cost can be significantly reduced as compared with the CFD. First, as shown in FIG.
([X '], [y'], [z ']); an orthogonal coordinate system in which the [z'] axis intersects with the vicinity of the center of the liquid surface at a substantially right angle.
Is set, the circumferential coordinate φ is set around the [z ′] axis, and the section from 0 to 360 degrees of φ is divided into N sections. Note that φ in the text is
[Outside 4]

, But represent the same thing. Each section
φ_j−0.5Δφ ≦ φ ≦ φ_j+ 0.5Δφ

, A tangent is drawn to the tank wall surface 4 at the intersection of the liquid level M and the tank wall surface 4 (plane φ = φ_jThe spherical coordinates ORθφ are set with the intersection of the tangent line and the [z ′] axis as the origin (the suffix j representing the circumferentially divided section is omitted to avoid complicating the formula). FIG. 3 shows an example of such spherical coordinate setting for two values of j. Using such spherical coordinates, the liquid surface M at the time of static equilibrium, the vibrating liquid surface F, and the tank wall surface W are defined.
M: R = R_M(Θ), F: R = R_F(Θ, φ, t) = R_M(Θ) + ζ (θ, φ, t), W: R = R_W(Θ) ... (2a, b, c)
It is expressed as here
R_M(Θ): R coordinate of liquid surface at static equilibrium
R_W(Θ): R coordinate of tank wall
ζ (θ, φ, t): Vibration displacement of liquid surface
[0038]
The expression such as equation (2) has the following advantages. In other words, the kinematic suitability that the liquid level displacement should follow (contact) the tank wall surface at the tank wall surface is satisfied by setting only one direction component (R component) of the displacement vector even if the tank wall surface is curved. Yes, and geometric handling is convenient.
[0039]
One thing to note is that the origin O in FIG. 3 jumps from the upper side of the tank to the lower side at φ, and the liquid level displacement で defined in the + R direction changes discontinuously at such φ. That is. In response to this, a parameter ε that takes -1, 1 is introduced depending on whether the origin is on the upper side or the lower side of the tank, and 符号 is inverted, and −εζ (from the liquid region to the outside) Consideration is given to developing the liquid surface displacement) as a continuous function using a characteristic function (equation (18)).
[0040]
Two coordinate systems (see Figure 3)
([X '], [y'], [z ']): spherical coordinate reference coordinate system
([X], [y], [z]): Tank reference coordinate system
Is generally given by the origin shift and rotation by Euler angles (FIG. 4):
(Equation 1)

Here, an example of the matrix component corresponding to the rotation is
E₁₁= Cosφ_Ecos θ_EcosΨ_E-Sinφ_EsinΨ_E,
E₁₂= Cosφ_Ecos θ_EsinΨ_E-Sinφ_EsinΨ_E, E_Thirteen= Cosφ_Esin θ_E  … (3b)
(Φ_E, Θ_E, Ψ_E): Euler angle
[0041]
[II] Equation of motion of liquid
The equation of motion of the liquid is given by the following equation.
(Δv_x) / (Δt) -2Ωv_y+ 1 / ρ (δP) / (δx) = 0, (δv_y) / (Δt) + 2Ωv_x+ 1 / ρ (δP) / (δy) = 0, (δv_z) / (Δt) + 1 / ρ (δP) / (δz) = 0 (4a, b, c)
here
P = p−ρgz− (1/2) ρΩ²(X²+ Y²…… (5)
v_x, V_y, V_z: Flow velocity component
p: pressure
ρ: density of liquid
Note that δ in the text is
[Outside 5]

, But represent the same thing.
[0042]
The equation of motion (4) is converted into the spherical coordinates introduced in step [I], and the following equation is obtained in consideration of the inertial force due to the excitation acceleration.
[0043]
(Equation 2)

here

[0044]
[III] Static liquid surface shape
Prior to the sloshing analysis, it is necessary to obtain a static liquid surface shape determined by a static equilibrium condition between centrifugal force and gravity. In the equation of motion (6), the flow velocity and the excitation acceleration are set to 0, and the pressure p is changed to the static pressure p._stAnd reduce it to the equilibrium equation. By integrating this equation, p = C₁(C₁Is the integration constant)
p_st= Ρ [gz + (1/2) Ω²(X²+ Y²)] + C₁  … (8)
Get. p_stIs an unknown liquid level z = z_stFrom the condition that it becomes 0 in (x, y), the function form z representing the liquid surface shape_st(X, y) is determined:
z = z_st(X, y) = − (Ω²/ 2g) (x²+ Y²)-(C₁/ Ρg)… (9)
Constant C₁Is determined from the liquid filling rate of the tank. From equation (9), the static liquid level is a paraboloid of revolution.
[0045]
[IV] Variational principle under gravity and centrifugal force
In equation (6), the pressure p is expressed as follows:
p = p_st+ P_slo+ P_imp  … (10)
here
p_st: Static pressure determined in step [III].
p_slo: Slosh pressure (unknown) generated by vibration of the liquid surface.
p_imp: Impact pressure. It is caused by inertia when the liquid moves rigidly as a unit due to the forced acceleration of the tank, and is given by the following equation.
(Equation 3)

Here, from the condition of the average pressure 0 on the liquid level M,
(Equation 4)

[0046]
Substituting equation (10) into equation of motion (6) and considering the static relation (8), omitting Coriolis acceleration
(Δ / δt) ｛v_R, V_θ, V_φ｝^T= -Grad (p_slo/ Ρ)… (13)
Slosh pressure and
(Δ / δt) Φ = −p_slo/Ρ...(14)
There exists a velocity potential Φ having the relationship Therefore, the variational principle expressed by the velocity potential [Equation (19) in Reference (1), Expression (18) in Reference (2)] is applied, and furthermore, Reference (3) p. Sloshing pressure p instead of velocity potential Φ following 600_sloAnd fluid displacement component
(Equation 5)

The partial integration with respect to time is performed by Then, normal vectors and the like appearing in the variation principle are displayed in spherical coordinates, and the liquid surface boundary conditions are linearly approximated to obtain the following equation.
[0047]
(Equation 6)

(F₄(Θ, φ), f₅(Θ, φ): Function determined by tank position, notation omitted)
[0048]
[V] Sloshing analysis under gravity and centrifugal force
This variation principle is converted into a mode equation by the Galerkin method. The spherical coordinates introduced in step [I] for analytically displaying the solution are effective. The solution of the pressure is found as the solution of the Laplace equation, and the result is then time integrated using equation (13) to derive a solution for the fluid displacement component:
(Equation 7)

Here, the characteristic function system appearing on the right side is as follows.
[0049]
(Equation 8)

here
L_l: Normalized constant
λ_mkAnd Θ_mk: K-th eigenvalue and eigenfunction determined as a solution to the next boundary value problem
Θ ″ + cot θΘ ″ + [λ− (m²/ Sin²θ)] Θ = 0 (22)
Θ '= 0 at θ = θ_max  … (23)
α_mk1: Equation α (α + 1) = λ_mkThe two roots of
α_mk1= 1/2 [-1− (1 + 4λ)_mk)^1/2], Α_mk2= 1/2 [-1+ (1 + 4λ)_mk)^1/2]
[0050]
Substituting the equations (17) and (18) into the equation of the variational principle (16) from which the forced acceleration term is omitted, the undetermined constant A_mklq, C_mkqTo derive a system of homogeneous linear equations for these undetermined constants. This is a form of eigenvalue problem
(-Ω²M + K) X = 0 (24)
(X: A_mklq, C_mkqColumn vector)
And then C_mkqOnly the eigenvalue problem can be reduced, and the eigenfrequency and the mode function are determined by solving this problem.
[0051]
Mechanical model 5 (see Fig. 2)
Two vibration directions: <x>, <y>
For each model, the lowest order mode among the modes that vibrate in the vibration direction is obtained for each model. Since these mechanical models 5 are obtained independently and similarly, only the vibration direction in the plane of FIG. 2 will be described: The solution is expressed by a mode function as follows.
[0052]
(Equation 9)

[0053]
Substituting equation (25-27) into the variation principle (16) and taking variation on the mode coordinates q (t), leads to a mode equation of the form:
(Equation 10)

The dynamic forces and moments that the tank receives from the liquid, calculated by integrating the dynamic pressure at the tank wall, can be expressed as:
(Equation 11)

(A, B, C, and D are constants)
Sine wave excitation of mode equation (28)
(Equation 12)

Is obtained and substituted into Equation (29), the frequency response of the force and moment to the excitation acceleration is obtained as follows.
[0054]
(Equation 13)

[0055]
[VI] Mechanical model creation under gravity and centrifugal force
The following equation is obtained by calculating the slosh mass motion equation, the force and the moment acting on the tank when the mechanical model 5 in FIG. 2 receives the similar vibration (30).
[0056]
[Equation 14]

here

[0057]
From the equations (32) and (33), the response of the mechanical model to the sinusoidal excitation (30) of the force and moment is calculated as follows.
(Equation 15)

here
ω_{<X>, mech}= [(K_{1 <x>}-M_{1 <x>}P_{0 <x>}) M_{1 <x>}  ]^1/2  … (36)
[0058]
The force and moment of the equation (31) are equal to the force and moment of the equation (35)._fFrom the same conditions, the constants of the mechanical model are determined as follows.
[0059]

[0060]
[VII] Calculation of separated natural frequency by Coriolis acceleration
The natural frequencies determined in step [V] are substantially equal in the two vibration directions (see FIG. 7), and one natural frequency ω_{0, slo}Can be considered This natural frequency has two natural frequencies ω due to the influence of Coriolis acceleration._{1, slo}And ω_{2, slo}To separate. In this step [VII], these values are predicted as the first step for correcting the mechanical model. Since the theory of Laplace equation cannot be used when there is Coriolis acceleration, a more general variation principle and the display of the solution are necessary. First, as the variation principle, the following equation obtained by variationally expressing the equation of motion (6) and adding it to equation (16) is used.
[0061]
(Equation 16)

here
[Equation 17]

[0062]
Next, the display of the solution will be considered. Since the Laplace equation cannot be used, we want to use a general orthogonal function for the part that depends on R. However, the upper and lower limit R of R_W, R_MIs not constant and varies as a function of θ. In such a case, the conventional frequent method is to divide the domain of θ into a number of sections, and express the solution for each of the sections by locally constant discontinuous eigenvalues and independent generalized coordinates, In this method, these generalized coordinates are determined from the continuous condition at the boundary with the adjacent section. Unlike this method, this evaluation method
(Equation 18)

Represents the solution as follows, and in cos corresponding to the eigenvalue
π (l-1) / (R_w-R_M…… (43)
Is treated as a continuous function, the continuity at the boundary with the adjacent section is satisfied by considering its derivative, and a common generalized coordinate is provided throughout the domain of θ.
a_mklq, B_mklq, C_mklq, D_mklq, E_mkq
Reduces the number of degrees of freedom, computation time, and cost. Substituting equations (41) and (42) into equation (40), and performing a variation calculation on generalized coordinates,
(-Ω²M_s+ IωC_s+ K_s) X_s= F_s  … (44)
A matrix equation of the form here
M_s, K_s: Mass and stiffness matrix
C_s: Inverse symmetric matrix derived from Coriolis acceleration term
X_s: A column vector with generalized coordinates arranged
[0063]
When this is solved and the frequency at which the resonance peak appears is found, the fundamental natural frequency of the sloshing is a value ω divided into two by Coriolis acceleration._{1, slo}And ω_{2, slo}I understand.
[0064]
[VIII] Modified mechanical model
One square m₁Is connected to the tank by springs 6 having different spring constants in the two directions of excitation, and the tank receives a spin around the z-axis.
[Equation 19]

here
k_{1 <x>}, K_{1 <y>}: Spring constant in two directions of excitation
u, v: dynamic displacement in two directions of excitation
[0065]
Based on this, the mechanical model 5 obtained by modifying the mechanical model 5 in step [VI] (expression (32) and an expression similarly derived for the other excitation direction, omitted) is expressed as follows.
[0066]
(Equation 20)

here
m_{1 <x>}, M_{1 <y>}: The value obtained in step [VI] with the inertial mass relating to the movement in the <x> and <y> directions.
c: a constant that controls the degree of natural frequency separation caused by the coupled term due to Coriolis acceleration;
k '_{1 <x>}, K '_{1 <y>}: Modified spring constant for adjusting the value of the separated natural frequency
[0067]
And the forces and moments are given by:
(Equation 21)

[0068]
The frequency equation of equation (46) is

[0069]
Ω in equation (48)²Imposing a condition that the mean and difference of the two roots with respect to the square of the separated natural frequency obtained in step [VII] are equal to the difference, respectively.
(−B) / (2A) = (ω_{1, slo} ²+ Ω_{2, slo} ²) / 2, ((B²-4AC)^1/2) / A = ω_{2, slo} ²−ω_{1, slo} ²  ... (50a, b)
Eliminating B from equations (50a) and (50b)
C / A = ω_{1, slo} ²ω_{2, slo} ²  … (51)
And substituting equation (49) gives:
[0070]
ω_{1, slo} ²ω_{2, slo} ²= (K '_{1 <x>}-M_{1 <x>}Ω²cos²<Γ>) / m_{1 <x>}・ (K '_{1 <y>}-M_{1 <y>}Ω²) / M_{1 <y>}  … (52)
[0071]
The first and second factors on the right side of equation (52) are products ω_{1, slo}ω_{2, slo}K 'to be equal to_{1 <x>}, K '_{1 <y>}Is determined. It will be confirmed in the next section that the frequency response of the mechanical model 5 obtained in this manner well represents the resonance behavior of the frequency response of sloshing.
[0072]
2. Application example
First, the parameters of the mechanical model in step [VI] were calculated for the case where there was no spin due to gravity only. The result is shown in FIG.
[0073]
FIG. 5A shows the natural frequency ω._{1 <x>}/ 2π, ω_{1 <y>}/ 2π (Hz),
(B) is slosh mass m_{1 <x>}, M_{1 <y>}(Kg),
(C) is slosh mass m_{1 <x>}, M_{1 <y>}(Total mass of liquid m_liquidDimensionless value),
(D) is the mounting position l of the slosh mass_{1 <x>}, L_{1 <y>}(Dimension-less value with spherical part radius),
(E) is the mounting position l of the fixed mass._{0 <x>}, L_{0 <y>}(The value obtained by dimensionlessizing the radius of the spherical portion) (under gravity control; solid line and dashed line indicate values for <x> and <y> excitation).
[A = 0.25m, θ_C= 40deg, γ = 43deg, L_cent= 0.5m, H_cent= 0m, α = 0, Ω = 0rpm, g = 2m / s², Ρ = 1009 kg / m³;
Slope <γ> = 0 of force reference coordinates (<x>, <y>, <z>), and (x ′, y ′, z ′) coordinates (L_cent, 0, H_cent+ A (see FIG. 2);
Slope (φ) of spherical coordinate reference coordinates ([x ′], [y ′], [z ′])_E, Θ_E, Ψ_E) = (0, −38 °, 0), [x], [y], [z]) coordinates (0.14, 0, 0.17) m of the origin (see FIGS. 3 and 4)]
[0074]
It should be noted that the slosh mass does not increase monotonically with increasing liquid filling rate, but peaks at an intermediate filling rate. The reason for this is that as the filling rate approaches 100%, the area of the liquid level becomes smaller and the constraint becomes stronger, while at a low filling rate, the slosh mass becomes smaller because the total liquid amount is smaller. As a result, the slosh mass becomes maximum at a slightly higher filling ratio than the middle.
[0075]
This calculation method has been devised for the case where the liquid shape at the time of static equilibrium is non-axisymmetric, but for verification, the conical part is made smaller (the vertex angle of the conical part is made larger) and the The parameters of the mechanical model were calculated for the case where the distance was approached, and the results are shown in FIG. 6 in comparison with the past literature (4).
[0076]
FIG. 6A shows a dimensionless natural frequency ω._{1 <x>}/ (G / a)^1/2, Ω_{1 <y>}/ (G / a)^1/2,
(B) is slosh mass m_{1 <x>}, M_{1 <y>}(Total mass of liquid m_liquidDimensionless value),
(C) is the mounting position l of the slosh mass_{1 <x>}, L_{1 <y>}(Dimension-less value with spherical part radius),
(D) is the mounting position l of the fixed mass_{0 <x>}, L_{0 <y>}(Value converted to dimensionless by the radius of the spherical portion) (verified by a spherical tank under gravity control, verified so that it could not be expressed by another curve with the result of reference (4) p. 13; solid line, chain line <X>, <y> values for excitation).
[A = 0.25m, θ_C= 70 deg, γ = 22 deg, L_cent= 0.5m, H_cent= 0m, α = 0, Ω = 0rpm, g = 2m / s², Ρ = 1009 kg / m³;
Slope <γ> = 0 of force reference coordinates (<x>, <y>, <z>), and (x ′, y ′, z ′) coordinates (L_cent, 0, H_cent+ A) (see FIG. 2);
Slope (φ) of spherical coordinate reference coordinates ([x ′], [y ′], [z ′])_E, Θ_E, Ψ_E) = (0, −17 °, 0), [x], [y], [z]) coordinates (0.07, 0, 0.01) m of origin (see FIGS. 3 and 4)]
[0077]
It has been confirmed that the agreement with the result of reference (4) is very good.
[0078]
FIG. 7 shows the calculation results of the parameters of the mechanical model when there is a spin of the present subject matter.
[0079]
FIG. 7A shows the natural frequency ω._{1 <x>}/ 2π, ω_{1 <y>}/ 2π (Hz),
(B) is slosh mass m_{1 <x>}, M_{1 <y>}(Kg),
(C) is slosh mass m_{1 <x>}, M_{1 <y>}(Total mass of liquid m_liquidDimensionless value),
(D) is the mounting position l of the slosh mass_{1 <x>}, L_{1 <y>}(Dimension-less value with spherical part radius),
(E) is the mounting position l of the fixed mass._{0 <x>}, L_{0 <y>}(Values rendered dimensionless by the radius of the spherical portion) (solid lines and dashed lines indicate values for <x> and <y> excitation).
[A = 0.25m, θ_C= 40deg, γ = 43deg, L_cent= 0.5m, H_cent= 0m, α = 0, Ω = 20rpm, g = 2m / s², Ρ = 1009 kg / m³;
Force reference coordinates (<x>, <y>, <z>) and spherical reference coordinates ([x '], [y'], [z ']) are tank target coordinates ([x], [y], [Z]) (see FIGS. 2 and 3)]
[0080]
The gradient of the liquid surface caused by spinning is at the position where the liquid surface intersects the center line of the tank.
dz '/ dx'
Is about -0.8 to -1.3, which is considerably large due to small gravity, and the separation of the natural frequency by Coriolis acceleration is also remarkable as shown below.
[0081]
8 and 9 show that the frequency response of the sloshing force and moment obtained by the analysis of step [VII] can be approximately represented by the modified mechanical model of step [VIII].
[0082]
FIG. 8
(Equation 22)

FIG. 9 shows the frequency response of force and moment to the liquid [liquid filling ratio 48%; solid line, mechanical model; dotted line, sloshing system; other parameters are the same as FIG. 7].
(Equation 23)

And the frequency response of the force and moment [liquid filling ratio 48%; solid line, mechanical model; dotted line, sloshing system; other parameters are the same as in FIG. 7]. The representative values of the intermediate filling rates, which are important due to the large slosh mass, are shown. In particular, the coincidence in the band between the two separated natural frequencies is good, and the coincidence slightly worse outside the two natural frequencies is because the calculation result of the sloshing system includes the contribution of the adjacent mode. (There is also a mode called internal vibration on the low frequency side). In particular, it should be noted that the following occurs due to coupling of vibrations in the two excitation directions due to Coriolis acceleration.
[0083]
A force in the <y> direction and a moment in the <x> direction are generated even in the <x> direction (these are not generated without Coriolis acceleration)
A force in the <x> direction and a moment in the <y> direction are also generated for the <y> direction excitation (these do not occur without Coriolis acceleration)
[0084]
The phase changes of the force and moment components caused by these couplings are the same in the modified mechanical model and the sloshing system.
[0085]
Literature
[1] Utsumi, M .; , 1998, "Low-gravity Propellant Slow Analysis Using Spherical Coordinates," Journal of Fluids and Structures, 12, pp. 57-83.
[2] Utsumi, M .; , 2000, "Low-gravity Slowing in an Axis Mechanical Container Excited in the Axial Direction," ASME Journal of Applied Mechanics, 67. 344-354.
[3] Utsumi, M .; , 2000, "Development of Mechanical Models for Propellant Sloshing in Teardrop Tanks," Journal of Space and Rockets, 37, p. 597-603.
[4] NASA SP-8809, NASA Space Vehicle Design Criteria (Structures), Propellant Slosh Loads, August 1968.
[0086]
It should be noted that the present invention is not limited only to the above-described embodiment, and it goes without saying that various changes can be made without departing from the spirit of the present invention.
[0087]
【The invention's effect】
According to the method for creating a mechanical model of sloshing in a teardrop-shaped tank in consideration of Coriolis acceleration according to the present invention, the following excellent effects can be obtained.
[0088]
When evaluating the frequency characteristics of sloshing to the vibration of a teardrop-shaped tank, the axis of the spherical coordinate intersects the liquid surface at an almost right angle near the center of the liquid surface so that it can be taken from anywhere in the tank cone, and the solution display Can be applied to non-potential flows, and the frequency characteristics of sloshing including Coriolis acceleration can be evaluated by an analytical method. Therefore, as compared with the conventional numerical method of the finite element method, the calculation time for analyzing the sloshing system, that is, for generating the design mass spring model is shortened, and the calculation cost is reduced.
[0089]
Therefore, based on the display of the solution, firstly, without considering the Coriolis acceleration, a design mass spring model that approximately represents the frequency characteristic in two vibration directions substantially parallel to the liquid surface and orthogonal to each other is separately created. Next, these design mass spring models are coupled so that the natural frequency can be separated by Coriolis acceleration, and the natural frequency of the sloshing separated by Coriolis acceleration is evaluated. Since the spring constant of each of the design mass spring models is modified so as to approximately represent the design mass spring model, a design mass spring model having a frequency characteristic very similar to the frequency characteristic of an actual sloshing system including Coriolis acceleration is obtained. Can be created.
[0090]
As described above, it is possible to create a design mass spring model having a frequency characteristic approximate to the frequency characteristic of sloshing including Coriolis acceleration by the sloshing system, thereby accurately evaluating the influence of sloshing on the satellite and controlling the attitude of the satellite. Performance can be greatly improved.
[Brief description of the drawings]
FIG. 1 is a conceptual diagram showing a teardrop-shaped tank to which the present invention is applied and a coordinate system thereof.
FIG. 2 is a conceptual diagram of force reference coordinates and a mechanical model.
FIG. 3 is a conceptual diagram of a method of setting spherical coordinates.
FIG. 4 is a conceptual diagram of coordinate conversion for determining the origin and axis of spherical coordinates.
FIGS. 5 (a), (b), (c), (d), and (e) are diagrams showing the results of calculating the parameters of the mechanical model in step VI in the case where there is no spin due to gravity only.
FIGS. 6 (a), (b), (c) and (d) show parameters calculated for a teardrop-shaped tank close to a spherical tank with a conical portion reduced and compared with past literature. FIG.
FIGS. 7 (a), (b), (c), (d), and (e) are diagrams showing calculation results when there is a spin of the main subject.
8 (a), (b), (c) and (d) show that the modified mechanical model of step VIII can approximately represent the frequency response of the force and moment due to sloshing obtained in the analysis of step VII. FIG.
9 (a), (b), (c) and (d) show that the modified mechanical model of step VIII can approximately represent the frequency response of the force and moment due to sloshing obtained in the analysis of step VII. FIG. 9 is a diagram shown under conditions different from those in FIG. 8.
FIG. 10 is a longitudinal sectional view for explaining the outline of a teardrop-shaped tank.
11 is a conceptual diagram illustrating a design mass spring model of the teardrop-shaped tank shown in FIG. 10;
[Explanation of symbols]
1 teardrop-shaped tank
4 Tank wall
5 Design mass spring model
6 Spring
M liquid level
m₁    trout
k spring constant

Claims (3)

A third coordinate axis substantially perpendicular to the liquid surface at substantially the center of the liquid surface of the teardrop-shaped tank whose central axis is inclined at a predetermined angle with respect to the spin axis; An orthogonal coordinate system having a second coordinate axis is set, and circumferential coordinates are set around the third coordinate axis, and a range from 0 ° to 360 of the circumferential coordinate is divided into a plurality of sections. A tangent is drawn to the tank wall surface at an intersection line between the liquid surface and the tank wall surface, spherical coordinates are set with the origin at the intersection of the tangent line and the third coordinate axis, and a first coordinate is set in the sloshing system of the teardrop-shaped tank. The frequency characteristics of the force in the first coordinate axis direction and the moment about the second coordinate axis that the tank receives when the tank is vibrated in the coordinate axis direction are determined, and the frequency characteristics in the first coordinate axis direction vibration are approximated. Create a design mass spring model In addition, the frequency characteristics of the force acting on the tank in the second coordinate axis direction and the moment about the first coordinate axis when the tank is vibrated in the second coordinate axis direction are determined, and the vibration in the second coordinate axis direction is obtained. In the meantime, a design mass spring model that can approximately represent the frequency characteristic at the time is created, while the natural frequency of the sloshing separated by Coriolis acceleration is evaluated, and each of the above-described respective natural frequencies is evaluated so as to approximately represent the evaluated separated natural frequency. A method for creating a mechanical model of sloshing in a teardrop-shaped tank in consideration of Coriolis acceleration, wherein a spring constant of a designed mass spring model is corrected. The Coriolis acceleration according to claim 1, wherein the evaluation of the natural frequency of the sloshing separated by the Coriolis acceleration is performed by introducing a variation principle applicable to a non-potential flow and a display of a solution. Method of creating a mechanical model of sloshing in a teardrop shaped tank. The correction of the spring constant of each of the design mass spring models is performed by separating the natural frequency into a coupled term in a vibration direction between the first coordinate axis and the second coordinate axis generated when Coriolis acceleration acts on each design mass spring model. 2. A teardrop shape in consideration of Coriolis acceleration according to claim 1, wherein a constant for controlling the degree is introduced, and a spring constant in each excitation direction is corrected in accordance with the evaluated natural frequency of the sloshing. How to create a mechanical model of sloshing in a tank.

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