円筒タンクにおけるポテンシャル流体と弾性容器との大変形動的不安定問題の解析

円筒タンクにおけるポテンシャル流体と弾性容器と

の大変形動的不安定問題の解析

著者

皆川 洋一

雑誌名

鹿児島大学工学部研究報告

巻

55

ページ

29-50

別言語のタイトル

Nonlinear Vibration Problem of Cylindrical

Tank with Water in Large Deformations

円筒タンクにおけるポテンシャル流体と弾性容器と

の大変形動的不安定問題の解析

著者

皆川 洋一

雑誌名

鹿児島大学工学部研究報告

巻

55

ページ

29-50

別言語のタイトル

Nonlinear Vibration Problem of Cylindrical

Tank with Water in Large Deformations

㣮ఽፉᄢቇᎿቇㇱ⎇ⓥႎ๔ ╙55 ภ㧔2013㧕  

౞╴࠲ࡦࠢߦ߅ߌࠆࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢᕈኈེߣߩ

ᄢᄌᒻേ⊛ਇ቟ቯ໧㗴ߩ⸃ᨆ

   ⊝Ꮉᵗ৻

*

Nonlinear Vibration Problem of Cylindrical Tank with Water in Large Deformations

Youichi MINAKAWA

It is reported that responses of liquid-filled tank on a shaking table showed some vibration modes that was contradiction to expectation of elementary tank theory. Then, author has been studying to analyze the response that might be caused by dynamic geometric nonlinear behavior, and showed a Lagrangian function that governed the interactive behavior between the potential fluid and elastic container in large deformations. Here, applying ALE(arbitrary Lagrangian-Eulerian Element) to the functional of a cylindrical tank in a three dimension, we propose a new procedure, analyze nonlinear responses of the system, and demonstrate the effectiveness of the method.

Keywords :Potential Fluid, Lagrangian of interaction between fluid and container, Nonlinear response of elastic tank, Sub-harmonic response

 ߪߓ߼ߦ  ᶧ૕ߣᒢᕈኈེߩㅪᚑ໧㗴ߪ㧘ᶧ૕ߩ⾰㊂ലᨐࠍ ᒢᕈኈེߦઃടߔࠆቯᑼൻ߇_{1950 ᐕઍ߹ߢ↪޿ࠄ} ࠇߡ޿ߚ㧚ߘߩᓟ㧘ࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢᕈኈེߩ ᓸዊᄌᒻ႐ߦ߅ߌࠆㅪᚑ໧㗴ߩ᳢㑐ᢙ߇㐿⊒ߐࠇߚ㧚 ߎࠇߦၮߠߊ✢ᒻℂ⺰߇ቯᑼൻߐࠇ㧘Ꮏቇ໧㗴ߦㆡ ↪ߐࠇ㧘᦭↪ߥ⍮⷗3,4,5,6)߇ᓧࠄࠇߡ޿ࠆ㧚 J.C.Luke1)_{ߪࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߩ࿶ജᑼࠍᵹ૕႐} 㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋㨋                        ᐕ  ᦬  ᣣฃℂ * ᑪ▽ቇኾ᡹ ߢⓍಽߔࠆ㑐ᢙ߇ߎߩᵹ૕ߩㆇേࠍᡰ㈩ߔࠆ෩ኒߥ Lagrange 㑐ᢙߣߥࠆߎߣࠍ␜ߒߚ㧚ߎࠇࠍ೑↪ߒߡ㧘 ೰ߥ࠲ࡦࠢߦ౉ߞߚᶧ૕ߩ෩ኒߥᝄേ໧㗴߇⸃ᨆߐ ࠇ㧘ታ㛎୯ߣߩᢛวᕈ߇ႎ๔ߐࠇߡ޿ࠆ㧚 ⪺⠪ߪ೨ㅀߒߚ_{Lagrange 㑐ᢙࠍࡐ࠹ࡦࠪࡖ࡞ࠛࡀ} ࡞ࠡߣߒߡ㧘ࡂࡒ࡞࠻ࡦߩේℂߦขࠅㄟߺ㧘ኈེߩ ᒢᕈᄌ૏ߦ઻߁ᵹ૕႐ߩᄌᒻࠍ⠨ᘦߔࠆߣ㧘ࡐ࠹ࡦ ࠪࡖ࡞ᵹ૕ߣᒢᕈኈེߩᄢᄌᒻേ⊛ㅪᚑ໧㗴ߦ߅ߌ ࠆ෩ኒߥ᳢㑐ᢙ7,8,9)ߦߥࠆߎߣࠍ␜ߒߚ㧚 ᵹ૕ߣᒢᕈኈེߩㅪᚑ໧㗴ࠍᄢᄌᒻߩਅߢ⸃ᨆߔ ࠆߣ߈㧘ᵹ૕ߣ⒖േߔࠆႺ⇇㧘ߔߥࠊߜㆇേߔࠆኈ

ེߣߩᢛวᕈࠍᜂ଻ߔࠆಣℂ߇ᔅⷐߣߥࠆ㧚᦭㒢ⷐ ⚛ᴺࠍ೑↪ߔࠆߣ㧘ߎߩ᧦ઙࠍචಽߥ♖ᐲߢḩ⿷ߔ ࠆߎߣߦ࿎㔍߇઻߁㧚߹ߚ㧘ࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߩ࿶ ജⓍಽߪ_{Euler ߩᣇᴺߢ⴫␜ߐࠇ㧘ᒢᕈኈེߪㅢᏱ} Lagrange ߩᣇᴺࠍ↪޿ߡ⴫␜ߐࠇࠆߩߢ㧘ᵹ૕ߣᒢ ᕈኈེߩ⋧੕૞↪㕙߇ᄢᄌᒻᤨߦᄌᒻ㨯⒖േߔࠆߎ ߣࠍ⊛⏕ߦ⴫⃻ߔࠆߎߣ߇㔍ߒߊ㧘቟ቯߥᔕ╵ࠍᓧ ࠆߎߣߪ࿎㔍ߢ޽ߞߚ㧚 ౞╴࠲ࡦࠢߩᶋ߈ደᩮߦ࿾㔡ⵍኂ߇⊒↢ߒ㧘ߎߩ ደᩮߩࠬࡠ࠶ࠪࡦࠣᝄേᤨߩᵄ㜞ߩ㕖✢ᒻ⸃ᨆ5,6) ߇ⴕࠊࠇߡ޿ࠆ㧚ߒ߆ߒߥ߇ࠄ㧘ᵹ૕ߣᄌᒻߔࠆኈ ེߩᄢᄌᒻ႐ߦ߅ߌࠆ⒖േႺ⇇໧㗴ߪㆡᱜߦಣℂߐ ࠇߚᚻᴺ߇೑↪ߐࠇߡ޿ࠆ⸶ߢߪߥ޿㧚 1990 ᐕઍߦ౉ߞߡ㧘⒖േߔࠆෳᾖᐳᮡࠍዉ౉ߔ ࠆ_{ALE㧔Arbitrary Lagrangian-Eulerian㧕ⷐ⚛}23)߇㐿 ⊒ߐࠇ㧘ᵹ૕ߩㆇേࠍᵹ૕☸ሶߩㆇേߣߪ⁛┙ߒߚ ᐳᮡ♽ࠍ↪޿ߡ⸘᷹ߔࠆߎߣ߇น⢻ߣߥߞߚ㧚ߎࠇ ࠍ೑↪ߔࠇ߫㧘⒖േᄌᒻߔࠆႺ⇇ߩ߽ߣߢ㧘☼ᕈ㧘 ࿶❗ᕈ╬ࠍ␜ߔ৻⥸ߩᵹ૕ߩ␜ߔᄙ᭽ߥ᜼േࠍ⸃ᨆ ߔࠆ੐߇น⢻ߣߥࠆ㧚 ᵹ૕ࠍ㕖☼ᕈ㧘᷵ߥߒߩࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߦ㒢ቯ ߔࠇ߫㧘⥄↱⴫㕙ࠍᜬߟᵹ૕ߣᒢᕈ૕ߩ⋧੕૞↪໧ 㗴ࠍᡰ㈩ߔࠆ਄⸥ߩ᳢㑐ᢙ߇ሽ࿷ߔࠆߩߢ㧘᦭㒢ⷐ ⚛ᴺࠍㆡ↪ߒ㧘㔌ᢔ♽ߩ႐ߩᣇ⒟ᑼࠍኈᤃߦ⺃ዉߔ ࠆߎߣ߇ߢ߈ࠆ㧚ߎߎߢߪ㧘ㆫ⒖ᐳᮡ♽ࠍ೑↪ߒ㧘 ᵹ૕߳⒖േ▵ὐࠍዉ౉ߒ㧘౞╴࠲ࡦࠢߦ߅ߌࠆࡐ࠹ ࡦࠪࡖ࡞ᵹ૕ߣᒢᕈኈེߩᄢᄌᒻേ⊛ㅪᚑ໧㗴ߦ߅ ߌࠆ෩ኒߥ᳢㑐ᢙ߳᦭㒢ⷐ⚛ᴺࠍㆡ↪ߒ㧘㔌ᢔൻߐ ࠇߚ႐ߩᣇ⒟ᑼߩ⺃ዉ㧘߅ࠃ߮㕖✢ᒻᔕ╵ߩᢙ୯⸃ ᨆࠍⴕ޿㧘ᚻᴺߩ᦭ὑᕈࠍታ⸽ߔࠆ㧚       ࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣኈེߩㅪᚑ႐ߩ᳢㑐ᢙ ߣ㔌ᢔൻᚻᴺ ⥄↱⴫㕙ࠍ᦭ߔࠆࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢᕈኈེ߇㧘 ㅦᐲ ߢㆇേߔࠆၮ⋚ߩ਄ߦ࿷ࠆ♽ࠍ⠨߃ࠆ㧚ၮ⋚ ߩ਄ߦ┙ߟⷰኤ⠪߇᷹ⷰߔࠆᵹ૕߳ㅦᐲࡐ࠹ࡦࠪࡖ ࡞ 㧘߅ࠃ߮ෳᾖᐳᮡߩ⒖േㅦᐲ ࠍዉ౉ߔࠆߣ㧘 ⪺⠪߇␜ߒߚࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢᕈኈེߩᄢᄌᒻ േ⊛ㅪᚑ႐ߦ߅ߌࠆ᳢㑐ᢙ 7,16,19)ߪᰴᑼߩ ࠃ߁ߦ⴫ߐࠇࠆ㧚 1 0 0[ L( , ) { ( / 2- )ˆ ( - )} t ALE m _{t V} L f I U M ’M ’M  ·  z z rdxd dzT 

³ ³³³

  u v v r g K  0 { ( ) ( )} ] S E s A h · 2 3 r dxd dt 

_³³

U u u /  v  u Tޓ     ₍₁₎ ߎߎߦ㧘

M

_{:♽ߣห৻ߩၮ⋚߆ࠄ᷹ⷰߒߚᵹ૕ߩㅦ} ᐲࡐ࠹ࡦࠪࡖ࡞_{,v :ၮ⋚ߩㅦᐲࡌࠢ࠻࡞,0 Ș:⥄↱⴫㕙} ߩᵄ㜞ࡌࠢ࠻࡞_, u: ᒢᕈኈེߩᄌ૏ࡌࠢ࠻࡞, r:ᄌ ᒻᓟߩ૏⟎ࡌࠢ࠻࡞_, 3 _{:ᒢᕈኈེߩࡐ࠹ࡦࠪࡖ࡞} ࠛࡀ࡞ࠡઃ㍳_1,V_L:ᵹ૕㗔ၞ, UL:ᵹ૕⾰㊂ኒᐲ, S A :ኈེ⴫㕙Ⓧ, UE:ኈེ⾰㊂ኒᐲ, h:ኈེߩෘߐg: ㊀ജടㅦᐲˆv :ㆫ⒖ᐳᮡㅦᐲ, r :౞╴ࠪࠚ࡞ඨᓘࠍ_s ᗧ๧ߔࠆ㧚ᵹ૕ߪ_{Euler ⴫␜ߐࠇ㧚ኈེߪ Lagrange} ⴫␜ߐࠇߡ޿ࠆ㧚 ♽ߦฝᚻ♽ߩ౞╴ᐳᮡ♽₍r, ,T z)ࠍዉ౉ߒ㧘z ゲߩ ේὐࠍ࠲ࡦࠢᐩߩਛᔃߣߒ㧘ㅒ㋦⋥ᣇะࠍᱜߣߔࠆ㧚 (1)ᑼߩ╙ 1 ᄌಽࠍ▚ቯߒ㧘ᰴᑼࠍᓧࠆ㧚 0 1 ₂ 0 0 ( , ) 0 0 ˆ [ ( ) ˆ ˆ ( ) ( ) ˆ { ( / 2 ) } ˆ { ( / 2 ) ( ) } L f i f i L L t ALE m _{t V S} L L z L f L L i S S L z L f S t L f z L i S I rdrd dz dS dS dS · g dS · z z dS U G U G G G G G G M’ M T M ’M U M ’M K U M ’M U K M ’M ’M K U M ’M ’M G                      

³ ³³³

³³

³³

³³

³³

³³

u v n v i n v u n v v r i n u v v r i n       K g 0 { ( ) / } ] (2) S t E s A U h   w3 w r dxd dtT 

³³

u u v  u ޓ ߎߎߦ㧘⸥ภn_{L:ฦႺ⇇ߦ߅ߌࠆᵹ૕ᄖะ߈ߩᴺ✢} ᣇะࡌࠢ࠻࡞㧘S_{f:ᵹ૕⥄↱⴫㕙㧘Si:ᵹ૕ߣኈེߩ⋧} ੕૞↪㕙㧘S_{0:࿕ቯႺ⇇㕙ࠍ␜ߔ㧚} (2)ᑼߩ╙ 1 ᄌಽ߆ࠄᓧࠄࠇࠆᵹ૕㗔ၞ㧘߅ࠃ߮ ฦႺ⇇ߦ߅ߌࠆႺ⇇᧦ઙᑼࠍᢛℂߔࠆ㧚 1) ᵹ૕ౝㇱ VLߩ᧦ઙ㧦’ M 0  (3) 2) ᵹ૕⥄↱⴫㕙 Sf㧦 ˆ ( ) | 0 f z L z z K ’M v Ki n       (4) 0 ˆ { ( / 2 ) } | 0 f L z z · · K M ’M ’M v v r gK n  (5) 3) ᵹ૕࿕ቯႺ⇇ S0㧦(’Mv nˆ) L 0    (6) 0 v M ˆv ( ) ALE m I M, uK

4) ᵹ૕ߣኈེߩ⋧੕૞↪㕙 Si㧦 ˆ (’M v u n) L 0 (7) 0 0 ˆ { ( / 2 ) ( )} | { ( ) / } L f L E · · z z h U M ’M ’M U 3         w w u v v r n u v u 0     g (8) (3)ᑼߪᵹ૕ߩၮ␆ᑼߢ޽ࠆ㧚ᵹ૕ߩㅦᐲߪ⒖േ ㅦᐲࠍᜬߟᐳᮡ♽ߢ᷹ⷰߐࠇߡ޿ࠆߩߢ㧘࿕ቯߐࠇ ߚᐳᮡߢ᷹ⷰߐࠇࠆㅦᐲߪ’Mvˆߣ⴫ߐࠇࠆ㧚 (4)ᑼߪ⥄↱⴫㕙ߦ߅޿ߡ㧘ᵹ૕ㅦᐲߣᵄ㜞ㅦᐲߩ ᵹ૕ᴺ✢ᣇะᚑಽ߇৻⥌ߔࠆߎߣ㧘_{(5)ᑼߪᵹ૕⴫} 㕙ߩᴺ✢ᣇะ߅޿ߡ㧘ᵹ૕࿶ജᚑಽ߇࠯ࡠߢ޽ࠆߎ ߣࠍ⴫ߔ㧚_{(6)ᑼߪ࿕ቯႺ⇇ߩᴺ✢ᣇะߩㅦᐲ߇࠯} ࡠߢ޽ࠆߎߣࠍ⴫ߔ㧚_{(7)ᑼߪߎߩႺ⇇ߩᴺ✢ᣇะ} ߦ߅޿ߡ㧘ᵹ૕ㅦᐲߣኈེߩㅦᐲ߇৻⥌ߔࠆߎߣࠍ ⴫ߔ㧚_{(8)ᑼߪኈེߩᴺ✢ᣇะߦᵹ૕࿶ജ߇૞↪ߒ} ߡ㧘ᒢᕈኈེߣߟࠅ޽߁ജቇ⊛᧦ઙࠍ⴫ߔ㧚᳢㑐ᢙ ߪ‛ℂ⊛ߦㆡಾߥߎࠇࠄၮ␆ᑼ㧘߅ࠃ߮Ⴚ⇇᧦ઙࠍ ਈ߃ࠆ㧚  ⷐ⚛ߩࡕ࠺࡞ൻ  ᵹ૕ߪ_{8 ▵ὐ 6 㕙૕ⷐ⚛ࠍណ↪ߔࠆ㧚ߎߩⷐ⚛ߪ} ⋧੕૞↪㕙S_iߦ߅޿ߡ㧘ኈེ㧔౞╴ࠪࠚ࡞㧕ߣធߔ ࠆ㧚౞╴ࠪࠚ࡞ⷐ⚛ߩ▵ὐߪೋᦼ⁁ᘒߦ߅޿ߡ㧘๟ ᣇะߦ╬ⷺᐲߢⷐ⚛ಽഀߐࠇ㧘_{z ᣇะߦߪᐳᮡ୯߇} ৻ቯࠍ᦭ߔࠆหᔃ౞⁁ߦሽ࿷ߔࠆ߽ߩߣ઒ቯߔࠆ㧚 ߔߥࠊߜ㧘౞ࠍ๟ᣇะߦ_Nc╬ಽ㧘޽ࠆ޿ߪඨ౞ࠍ 1 c N  ╬ಽߔࠆ_NcᧄߩᲣ✢ࠍ᦭ߔࠆࡕ࠺࡞ߣߥࠆ㧚 ⋧੕૞↪㕙਄ߩࠪࠚ࡞▵ὐߪߔߴߡߎߩᲣ✢਄ߦሽ ࿷ߔࠆ㧚⋧੕૞↪㕙਄ߩᵹ૕▵ὐ߽ᄌᒻ೨ᓟࠍㅢߓ ߡ㧘ߎߩᲣ✢਄ߦ޽ࠆ㧚ߎߩᵹ૕▵ὐߩೋᦼ_{z ᐳᮡ} ߪ㧘ห৻ᐳᮡ୯㧔౞╴ࠪࠚ࡞ߣߪ⇣ߥࠆ_{z ᐳᮡ୯㧕} ࠍᜬߟหᔃ౞ߦሽ࿷ߔࠆߣ઒ቯߔࠆ㧚ߎࠇࠄߩ઒ቯ ߪ◲නߩߚ߼ߩᛒ޿ߢ޽ࠅ㧘ઁߩㆬᛯ߽น⢻ߢ޽ࠆ㧚 ኈེߪ _{Lagrange ᐳᮡࠍ↪޿ߡ⴫ߐࠇߡ޿ࠆߩߢ㧘} ᄌᒻᓟߦ߅޿ߡ▵ὐ߇タߞߡ޿ࠆᲣ✢ߪ⋥✢ߢߪߥ ߊ৻⥸ߦⓨ㑆ᦛ✢ߣߥࠆ㧚ᵹ૕▵ὐ߽ߎߩⓨ㑆ᦛ✢ ਄ߦᏱߦሽ࿷ߒ㧘ㆫ⒖ᐳᮡ♽ࠍ᭴ᚑߔࠆ㧚ᵹ૕ߩㆇ േߪߎߩⓨ㑆ᦛ✢਄ߩὐߢ᷹ⷰߐࠇࠆ㧚 ߎߩᣇᴺߪኈེࠍ೑↪ߒߡ㧘⋧੕૞↪㕙਄ߩᵹ૕ ▵ὐߩ૏⟎ࠍቯ߼ࠆߩߢ㧘ᵹ૕▵ὐߦ߅޿ߡᵹ૕ߣ ኈེ߇ኒធߔࠆ᧦ઙࠍኈᤃߦዉ౉ߔࠆߎߣ߇ߢ߈ࠆ㧚  ⒖േႺ⇇ߩㆡว᧦ઙ ߹ߕ㧘ᵹ૕ߩㆇേࠍ᷹ቯߔࠆෳᾖᐳᮡࠍ᭴ᚑߔࠆ ᵹ૕▵ὐࠍㆇേߩ઀ᣇߦᔕߓߡ㧘_{5 ⒳㘃}16,19)ߦಽ㘃 ߔࠆ㧚ߘࠇࠄߪ㧘_{(1)ᵹ૕⥄↱⴫㕙}S_f ߣ⋧੕૞↪㕙 i Sߩਔ㕙ߦ࿷ࠆ▵ὐ(2 ⒖േႺ⇇㕙਄ߩ▵ὐ), (2)ᵹ ૕⥄↱⴫㕙਄ߩ▵ὐ㧘_{(3)⋧੕૞↪㕙਄ߩ▵ὐ㧘(4)} ⒖േߒߥ޿▵ὐ㧘߅ࠃ߮_{(5)ᵹ૕ߩౝㇱ▵ὐߢ޽ࠆ㧚}

 Fig.1 Fluid Node Move and Reference Coordinates

1) ⋧੕૞↪㕙਄ߩᵹ૕▵ὐߩ⒖േ  _{2 ⒖േႺ⇇㕙਄ߩᵹ૕▵ὐߦᵄ㜞߇↢ߓࠇ߫㧘⋧} ੕૞↪㕙ߪ㕙Ⓧ߇ᄌൻߔࠆ㧚_{Fig.1 ߦ߅ߌࠆ 2 ⒖േ} Ⴚ⇇㕙਄ߩ▵ὐk_fm (z ᐳᮡࠍz_fmߣߔࠆ_)ߩᵄ㜞ࠍ fm K ߣߔࠆߣ㧘ߎߩὐߩᵹ૕⒖േߪᵄ㜞ߣ㧘ߘߩ೔ ㆐ὐߩᒢᕈᄌ૏ߩ๺ߣߒߡ⴫ߐࠇࠆ㧚            ( , , ) ( , , ) ( , ) ( , ) fm s m fm fm s m fm m fm m fm m fm m r z r z s x ' T K T K T K I     c    r r r u i u (9) ߎߎߦ㧘ᷝሼfߪߘࠇߙࠇᵹ૕⥄↱⴫㕙㧘߅ࠃ߮ᷝ ሼm ߪ਄⸥ߒߚN_cᧄߩᲣ✢ߩ߭ߣߟࠍઍ⴫ߔࠆ⇟ ภߢ޽ࠅ㧘Უ✢T T_mߩ㊂ߢ޽ࠆߎߣࠍ␜ߔ㧚⋧੕ ૞↪㕙਄ߩᵄ㜞ߩᣇะi ߪ౞╴ࠪࠚ࡞Უ✢ߩᄌᒻᓟ_mc ߩធ✢ᣇะࡌࠢ࠻࡞㧘౞╴ᐳᮡ♽ߩᐳᮡ୯s_mK_fm㧘 m T ߪࠪࠚ࡞ߩⷐ⚛ᐳᮡ୯_xm㧘߅ࠃ߮I_m(ઃ㍳ 2)ߦ ኻᔕߔࠆ㧚_{(9)ᑼߪ 2 ⒖േႺ⇇㕙਄ߩᵹ૕▵ὐ߇ᄌ} ᒻᓟ߽ኈེ਄ߦሽ࿷ߔࠆߎߣࠍᜂ଻ߔࠆ㧚ኈེߩਛ ᄩ㕙ߣ⋧੕૞↪㕙ߪኈེෘߩ _{1/2 ⒟ᐲߩ〒㔌߇޽ࠆ㧚} ߎߩ〒㔌ߪዊߐ޿ߣ઒ቯߒߡήⷞߔࠆ㧚 z-axis r i _i[ [ P i Q j kf P _k fm kim kPQ

(1) SkewGrid and Points

z i Sf S_i (2) WaveHeight&Disp. f P K z-axis r i i[ z i P i Q j fm K fm

u

(3)Reference Co-ordinates z i f P K r i [ i P i Q j fm K fm

u

z-axis ' kfm ' kiQ

ห᭽ߦ㧘ห৻Უ✢਄ߩ⋧੕૞↪㕙਄ߦ޽ࠆᵹ૕ౝㇱ ▵ὐ_kim㧔_{z ᐳᮡࠍ}z_imߣߔࠆ㧕ߩ⒖േߪᰴᑼߩࠃ߁ ߦቯ߼ࠄࠇࠆ㧚 / ( ) ( , ) im fm zim zf m xm m 'r K i u c T        ₍₁₀₎ ߎߎߦ㧘xc_imߪᐳᮡ୯simKfmzim/zf ߦኻᔕߔࠆ౞╴ ࠪࠚ࡞ⷐ⚛ߩx ᐳᮡ୯ߢ޽ࠆ㧚xc_imߪᵄ㜞K_fmࠍ฽߻ ߩߢ㧘ᵹ૕▵ὐߩ⒖േ㊂ߪᵄ㜞ߩ㑐ᢙߣߥࠆ㧚 ᵄ㜞ߦࠃࠆ⒖േ㊂ߪ㧘ᵄ㜞K_fmࠍೋᦼᐳᮡߩz ୯ ࠍ೑↪ߒߚౝಽᲧߦᔕߓߡቯ߼㧘ߘߩὐߩࠪࠚ࡞ᄌ ૏ࠍㅊടߒߡቯ߼ࠆ㧚ߎߩᲣ✢਄ߦ޽ࠆೋᦼᐳᮡ im x ߩ▵ὐߦ߅ߌࠆኈེߩᄌ૏ߪ㧘ࠪࠚ࡞ᄌ૏ᚑಽ ࠍ↪޿ߡ⴫ߔߎߣ߇ߢ߈ࠆ㧚 ( , )im m ( , ) /im m s w xc T v xc T r u       ₍₁₁₎ ߎߎߦ㧘ᵹ૕ߣኈེߩㆡว᧦ઙߦ߅޿ߡ㧘౞╴ࠪࠚ ࡞ߩᲣ✢ᣇะߩᄌ૏ߪήⷞߔࠆ㧚ᵄ㜞ߩ㜞ߐߣᲧセ ߒߡ㧘ߎߩᄌ૏ߪዊߐ޿ߣ್ᢿߒߚ㧚 2) ⥄↱⴫㕙ߩᵹ૕▵ὐߩ⒖േ  ᵄ㜞ߪ㋦⋥ᣇะߦ⊒↢ߒ㧘ࠪࠚ࡞ߩᄌ૏ߪᵹ૕▵ ὐߩr_m0dr_sᐳᮡߣࠪࠚ࡞ඨᓘr_sߩౝಽᲧߦᔕߓߡౝ ㇱᵹ૕▵ὐߩ⒖േߦᓇ㗀ߔࠆߣ઒ቯߔࠆ㧚 / / ( )m (im s)( ( , )im m r ( , ) )im m fP fP zfP zf r r w x v x T 'r K i  c T i  c T i (12) ኈེߩ㋦⋥ᣇะߩᄌ૏ߪ㧘⋧੕૞↪㕙਄ߦߥ޿ᵹ ૕▵ὐߩ_{z ᣇะߦᓇ㗀ߒߥ޿ߣߔࠆ㧚ߎࠇߪ 2 ⒖േ} Ⴚ⇇਄ߩᵄ㜞ߣ㧘⥄↱⴫㕙ߩߺߩ਄ߦ࿷ࠆᵄ㜞ߩ㑐 ਈ߇⇣ߥࠆߎߣࠍᗧ๧ߔࠆ㧚ᧄ⺰ߢߪᵄ㜞ࡌࠢ࠻࡞ ో૕ࠍȘ ߣ⴫⃻ߔࠆ㧚ߎߩࡌࠢ࠻࡞ߩㇱಽ㓸วߢ_e ޽ࠆ _{2 ⒖േႺ⇇਄▵ὐߩᵄ㜞ࡌࠢ࠻࡞ࠍ}Ș ߣ⴫ߔ㧚_m m Ș ߪኈེᄌ૏ߩᄌᢙߣߥࠆ㧚 3) ౝㇱ▵ὐߩ⒖േ  ౝㇱ▵ὐߩ⒖േߪᰴᑼߩࠃ߁ߦቯ⟵ߔࠆ㧚ฦ▵ὐ ߣหߓr,Tᐳᮡࠍᜬߟ⥄↱⴫㕙਄ߩᵄ㜞ࠍฦ▵ὐߩ zᐳᮡߣߩౝಽᲧߢ㧘㋦⋥ᣇะߦ⒖േߒ㧘หߓT,z ᐳᮡࠍᜬߟࠪࠚ࡞ߩᄌ૏ᚑಽw v, ߪ▵ὐߩrᐳᮡߣ ࠪࠚ࡞ඨᓘr_sߩᲧߦᔕߓߡ⒖േߔࠆߣ઒ቯߔࠆ㧚 / / ( ) ( )z ( s) ( , )im m r ( , )im m f z zf r r w x v x PQ P PQ PQ T 'r K i  c T i  c T i (13)   ᐳᮡ♽ߣᭂᐳᮡ⴫␜  ౞╴࠲ࡦࠢࠍ⸃ᨆߔࠆߩߢ㧘౞ࠍᄙⷺᒻߦㄭૃߖ ߕߦ㧘౞ᩇᐳᮡࠍዉ౉ߒߚ_{8 ▵ὐߩ 6 㕙૕ᵹ૕ⷐ⚛} ࠍ೑↪ߔࠆ㧚ߎߩᵹ૕ⷐ⚛ߩᒻ⁁㑐ᢙNi( , , )[ [ [₁i ₂i ₃i ࠍ↪޿ࠆߣ㧘ᵹ૕ߩೋᦼᐳᮡߪ㧘ᰴᑼߩࠃ߁ߦ⴫ߐ ࠇࠆ㧚 8 8 1 1 3 1 1 3 1 1 8 1 1 3 1 ( , , ) , ( , , ) , ( , , ) i i i i i e i i i i i e i i i i i i i e i r N r N z N z [ [ [ T [ [ [ T [ [ [

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Nr Nș Nz (14) ߎߎߦ㧘Nk (1 [ [₁k ₁)(1[ [₂k ₂)(1[ [₃k ₃) / 8, ( k 1, ,8).  ߎࠇࠄߩᑼߦ㧘ฦ▵ὐߩฦᣇะߩೋᦼᐳᮡ୯ࠍਗ ߴߡ᭴ᚑߐࠇࠆࡌࠢ࠻࡞ߦᷝ߃ሼeࠍઃߒߡ⴫ߒߚ㧚 ౞ᩇᐳᮡߪz ゲ߇․⇣ὐߥߩߢ㧘ߎߩゲ਄ߩᐳᮡ୯ ࠍ೑↪ߔࠆߣ߈⧯ᐓߩ㈩ᘦ߇ᔅⷐߣߥࠆ㧚   ౞╴ࠪࠚ࡞ⷐ⚛  ౞╴ࠪࠚ࡞ࠍ㔌ᢔൻߔࠆⷐ⚛ߣߒߡ㧘_{1 ▵ὐ 5 ⥄} ↱ᐲ{ , , ,u v wk k k E Exk, Tk}㧘4 ▵ὐߩⷐ⚛(k 1, ,4)ࠍ೑ ↪ߔࠆ㧚ߎߎߦ㧘ᄌ૏ᚑಽu v w, , ߪߘࠇߙࠇᲣ✢ᣇ ะ㧘๟ᣇะ㧘߅ࠃ߮ᴺ✢ᣇะߩᄌ૏ߢ޽ࠅ㧘E Ex, T 㧔ઃ㍳_{1 ߦቯ⟵ߒߚ㧕ߪᲣ✢ᣇะ㧘ߘࠇߙࠇ๟ᣇะ} ߩ࿁ォࠍ⴫ߔ㧚౞╴ࠪࠚ࡞ߩ߭ߕߺ⴫⃻ߪ_Sanders ߦࠃࠆ೰૕ᄌᒻ21)ࠍ⠨ᘦߒߚ߭ߕߺࠍ೑↪ߔࠆ㧚 ᦭㒢ᄌᒻߩ౞╴ࠪࠚ࡞ߪᰴᑼࠍ೑↪ߔࠆ㧚 2 2 , / 2, ( , ) / / 2, , / , x ux x T vT w rs T u r vT s x x T H E  H  E  J  E E ,, , / , , ( , ) / x x x T T T rs Tx xT rs N E  N E  F E  E Z (15) ,, ( , ) / , ( , , / ) /(2 ) x wx T v wT rs vx u rT s rs E   E  Z   ߎߩၮ␆ᑼߩ⺃ዉ╬ࠍઃ㍳_{1 ߦ␜ߔ㧚} ౞╴ࠪࠚ࡞߳࿁ォࠪࠚ࡞ⷐ⚛ࠍ೑↪ߔࠆߣ㧘✢ᒻ ⸃ᨆߪᭂ߼ߡኈᤃߦߥࠆ㧚๟ᣇะߩ‛ℂ㊂ࠍࡈ࡯࡝ ࠛ⚖ᢙߦዷ㐿ߒߡ㧘ฦࡈ࡯࡝ࠛ⚖ᢙᰴᢙߩᣇ⒟ᑼߦ ಽ⸃ߒߡቯᑼൻߢ߈ࠆ㧚 ߒ߆ߒߥ߇ࠄ㧘ᧄ⺰ᢥߩቯᑼൻߢߪ౞╴ࠪࠚ࡞ⷐ ⚛ࠍㆡ↪ߔࠆߩߢ㧘೑↪ߒߡ޿ࠆⷐ⚛ߩ♖ᐲࠍᛠី ߔࠆᔅⷐ߇޽ࠆ㧚࿕᦭ᝄേᢙ⸃ᨆߦ߅޿ߡ㧘_{3 ⒳㘃} ߩⷐ⚛㧔_{Sabir ⷐ⚛㧘ਃᰴరᐔ᧼ⷐ⚛㧘߅ࠃ߮ Sabir}

ⷐ⚛ࠍୃᱜߒߚⷐ⚛㧕ࠍᢙ୯⸃ᨆ17)ߒߡ㧘ฦⷐ⚛ ߩ․ᕈࠍᛠីߒ㧘_{1 ⒳㘃ߩⷐ⚛ࠍណ↪ߒߚ㧚ߎߩⷐ} ⚛ߩ᭎ⷐࠍઃ㍳_{2 ߦ␜ߔ㧚ߎߩⷐ⚛ߪ೰૕ᄌᒻߦࠃ} ࠆ߭ߕߺ߇↢ߓߥ޿㧘߅ࠃ߮৻ቯ߭ߕߺߩ᧦ઙࠍḩ ߚߒߡ޿ࠆ㧚  ߎߩ࠲ࠗࡊߩࠪࠚ࡞ⷐ⚛ߩᴺ✢ᣇะᄌ૏ߪઍᢙ₃ ᰴ㑐ᢙߢᄌൻߔࠆ㧚ㄝਔ┵ߩ▵ὐ⥄↱ᐲߪᄌ૏ߣ࿁ ォߢ޽ࠆ߆ࠄ㧘࿁ォߪ_{2 ᰴ㑐ᢙએਅߩઍᢙ㑐ᢙߢ⴫} ߐࠇࠆߎߣ߇ㆡಾߢ޽ࠆ㧚ߎࠇࠍ⠨ᘦߒߡᄌ૏㑐ᢙ ߩ৻ㇱࠍୃᱜߒߚᄌ૏㑐ᢙࠍ೑↪ߔࠆ㧚  ▵ὐᄌ૏ࡌࠢ࠻࡞d_eࠍ೑↪ߔࠆߣ㧘౞╴ࠪࠚ࡞ⷐ ⚛ߩᄌ૏ࡌࠢ࠻࡞ߪᰴᑼߩࠃ߁ߦ⴫ߐࠇࠆ㧚 ( , ) ( , ) ( , ) s u u e s v e v e s w w e u x v x w x I I I NL NL d u NL d NL d NL NL d ­ ½ ª º ­ ½ ° ° « » ° ° ® ¾ _{« »} ® ¾ ° ° _{« »} ° ° ¯ ¿ ¬ ¼ ¯ ¿    ₍₁₆₎ ߎߎߦ㧘ⷐ⚛ᐳᮡ♽ߩIᐳᮡ୯ߣ౞╴ᐳᮡ♽ߩTᐳ ᮡ୯ߪ✢ᒻߩ㑐ଥࠍ᦭ߔࠆᐳᮡࡄ࡜ࡔ࡯࠲ߢ޽ࠆ㧚 㔌ᢔൻᚻᴺ16,19) 2.5.1 ᵹ૕▵ὐߩᄌᒻᓟߩ૏⟎  ᵹ૕▵ὐߩᄌᒻᓟߩ૏⟎ߪ_{2.2 ▵ߢቯ⟵ߐࠇࠆ㧚} ߎࠇࠄࠍ↪޿ߡ㧘ᵹ૕ⷐ⚛ߩ_{8 ὐߩ⒖േ㊂ࠍᵹ૕ⷐ} ⚛ߩᒻ⁁㑐ᢙߦઍ౉ߒߡ㧘⒖േ㊂ߪᰴᑼߩࠃ߁ߦࡑ ࠻࡝ࠢࠬ⴫␜ߐࠇࠆ㧚 ( ) ( ) r r e e e e d m e z z r z T T K ' ' 'T ' ­ ½ ª º ª º ° ° « » _« » _ ® _{¾ « » « »} ° _{° « » « »} ¯ ¿ ¬ ¼ ¬ ¼ X C r X Ș C d B d Ș B Ș d X C   ₍₁₇₎ ߎߎߦ㧘dž_eߪ⥄↱⴫㕙਄ߩᵄ㜞ࡌࠢ࠻࡞㧘d_eߪ౞ ╴ࠪࠚ࡞ߩ▵ὐᄌ૏ࡌࠢ࠻࡞ߢ޽ࠆ㧚_{(17)ᑼߦ߅޿} ߡ㧘BKߪdeߩ㑐ᢙߢ޽ࠅ㧘Bdߪdžeߩ㑐ᢙߣߥࠆ㧚 ⋧੕૞↪㕙਄ߦ_{4 ▵ὐࠍ᦭ߔࠆᵹ૕ⷐ⚛ߪ 2 ᧄߩ} Უ✢਄ߦߘࠇߙࠇ_{2 ୘ߩ▵ὐࠍᜬߟ㧚ߎࠇࠄ▵ὐߩ} ᄌᒻᓟߩ૏⟎ߪ_{2.2 ▵ߦ␜ߒߚᐳᮡ}s ࠍ↪޿ߡ␜ߐ_m ࠇࠆ㧚ߎߩᐳᮡ୯ߦኻᔕߔࠆ౞╴ࠪࠚ࡞ߩᄌ૏v w, ࠍቯ߼ࠆ㧚ߎߩኈེ▵ὐߪห৻ߩᲣ✢਄ߦ޽ࠅ㧘ߎ ߩ▵ὐࠍ᜽߻౞╴ࠪࠚ࡞ߩ_{2 ὐߩ▵ὐᄌ૏ࡌࠢ࠻࡞} ࠍ↪޿ߡቯ⟵ߔࠆߎߣ߇ߢ߈ࠆ㧚ኈེ਄ߦ޽ࠆᵹ૕ ⷐ⚛ߩ▵ὐߪ_{4 ୘޽ࠆ㧚ߎࠇࠄ▵ὐߦ߅ߌࠆኈེߩ} ᄌ૏ࠍ৻⟵⊛ߦቯ߼ࠆߚ߼ߦ㧘ᦨዊߢኈེߩ_{4 ▵ὐ㧘} ᦨᄢ_{8 ▵ὐߩ▵ὐᄌ૏ࡌࠢ࠻࡞߇ᔅⷐߣߥࠆ㧚}  ◲නߥ଀ࠍ␜ߔ㧚ೋᦼᐳᮡr₀ ( , , )r_s T_m z_im ߩ⋧੕ ૞↪㕙਄ߩ▵ὐߪ_{(10)ᑼ߆ࠄᰴᑼߩࠃ߁ߦ⴫ߐࠇࠆ㧚} 2 2 ( / ) /(1 ) ( ) 0 (1/ ) ( ) ( / ) /(1 ) 0 s im f x x fm w im e m s w im e im im f x fm r z z x r x z z z E E K T E K c ­  ½ ­ ½ ­ ½ ° ° ° °_{ }° _c ° ® ¾ ® ¾ ® ¾ ° ° ° _ ° ° ° ¯ ¿ ¯ ¿ ¯ ¿ L d r L d (18) ߎߎߦ㧘Ex w z_,z( )im 㧚 (12)ᑼ㧘߅ࠃ߮(13)ᑼ߽ห᭽ߦ⴫␜ߢ߈ࠆ㧚ߎࠇ ࠄࠍⴕ೉⴫␜ߒߡ㧘ᰴᑼࠍᓧࠆ㧚 0 ' 0 K e d e    r r r r B Ș B d       (19)  ߎߩᑼߪᵄ㜞㧘߅ࠃ߮ኈེ߇ᒢᕈᄌ૏ߒߚᄌᒻᓟ ߩᵹ૕▵ὐߩᐳᮡ૏⟎ࠍ⴫ߔ㧚 2.5.2 ᭂᐳᮡߣᵹ૕ⷐ⚛ߩ⴫␜  ᵹ૕ⷐ⚛ߩᒻ⁁㑐ᢙ㧘߅ࠃ߮ᵹ૕▵ὐߩㅦᐲࡐ࠹ ࡦࠪࡖ࡞㨯ࡌࠢ࠻࡞M_eࠍ↪޿ߡ㧘ㅦᐲࡐ࠹ࡦࠪࡖ࡞ M ߪᰴᑼߩࠃ߁ߦ⴫ߐࠇࠆ㧚 8 1 k k e k N = M

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M N          (20) M  ో૕ᐳᮡ♽ߣߒߡߩ౞╴ᐳᮡ( , , )rT z ߣ㧘ᒻ⁁㑐 ᢙࠍ↪޿ߚⷐ⚛ߩᐳᮡ( , , )[ [ [_{1 2 3} ߩ㑐ଥࠍ␜ߔ㧚 1 1 1 1 2 2 2 2 3 3 3 3 / / / / / / / ( ) / / / / / / / / r r z r r z z r z [ [ T [ [ [ T [ T [ [ [ [ T [ [ w w w w w w w w w w ­ ½ ­ ½ ª º °_{w w} ° °_{w w} °_{  w w}« _{w w w w} » ® ¾ ® ¾ _{« »} °_{w w} ° ° _{w w} ° _{«w w w w w w »} ¯ ¿ ¯ ¿ ¬ ¼ J J (21) ߎߎߦ㧘_{J ߪࡗࠦࡆࠕࡦⴕ೉㧚} (21)ᑼ߆ࠄ㧘ᰴᑼߩ⴫⃻ࠍᓧࠆ㧚 1 1 2 3 / / / ( ) grad / / / r r z [ T ’ [ [  w w w w ­ ½ ­ ½ °_{w w} ° °_{w w} ° ® ¾ ® ¾ ° _{w w} ° °_{w w} ° ¯ ¿ ¯ ¿ J     ₍₂₂₎  _(22)ᑼࠍMߦ૞↪ߐߖߡ㧘ᰴᑼࠍᓧࠆ㧚 1 1 2 3 / 1 grad / | | / e e e [ ’M M [ [  w w ­ ½ °_{w w} ° ® ¾ °_{w w} ° ¯ ¿ J J M M N A    ₍₂₃₎ ߎߎߦ㧘|J|ߪࡗࠦࡆࠕࡦߩⴕ೉ᑼߩ୯ߢ޽ࠆ㧚

 ෳᾖᐳᮡߩㅦᐲ_{ˆv ߪ(19)ᑼࠍᤨೞߢᓸಽߒߚᰴᑼ} ߢቯ⟵ߐࠇࠆ㧚 ˆ= v rޓ                ₍₂₄₎  _{(18),(19),(21)ᑼ╬ࠍ(1)ᑼ߳ઍ౉ߔࠆߣ㧘ᄌᒻᓟߩ} ᒻ⁁ߢ⹏ଔߐࠇ㧘㔌ᢔൻߐࠇߚ᳢㑐ᢙࠍᓧࠆ㧚 1 1 1 1 0 ₁ 1 1 1 0 0 ( ) [ { ( ) 1 _{( ( ) )}} 2 1 { ( ) ( )} ] 2 L s t ALE t t m e e e _t L e e e m t t e e f t E s A m I A · N N g z z d d d J h r dxd dt    M U M M M M [ [ [ U 3 T          ˜  ˜ 

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, d N J r v r J J u u u v u       K  (25) (24)ᑼߩෳᾖㅦᐲߪߔߴߡߩฦⷐ⚛▵ὐߩᄌ૏ߦ ଐሽߔࠆ㧚_{(25)ᑼߪ㧘ⷐ⚛ߩᄌᒻ೨ߩᐳᮡߦၮߠ޿} ߡ㧘ᄌᒻᓟߩᒻ⁁ߦ㑐ߒߡⓍಽߔࠆߎߣࠍ⴫ߔ㧚 (25)ᑼߩฦ㗄ߦኻߔࠆᄌಽᑼࠍ␜ߔ㧚  ࡗࠦࡆࠕࡦJߪࠬࠞ࡜࡯ߢ޽ࠅ㧘dž_e,Ș ,߅ࠃ߮_m e d ߩ㑐ᢙߢ޽ࠆ㧚ᄌᢙߦ㑐ߔࠆჇಽࠍ⴫␜ߔࠆ㧚 e m m d e J K ' D Ș' D Ș' D d'         (26) ߎߎߦ㧘 _t , _{m t} , _{d t} m e e J J J K w w w w w w D D D Ș Ș d (26.1)  Ae eM ߪࡌࠢ࠻࡞ߢ޽ࠅ㧘ฦᚑಽࠍห᭽ߦᄌᢙMe, e dž ,Șc,߅ࠃ߮deࠍ↪޿ߡᓸಽߒߡ㧘ᰴᑼࠍᓧࠆ㧚 ( ) + ( ) , e e e e e m m d e e e d e d e d e m m r d d K K K ' ' ' ' ' ' ' ' ' ' '      ޓ ޓ A A S Ș S Ș S d B Ș B Ș T d B d B d B Ș B B T M M  ₍₂₇₎ (26)㧘(27)ᑼࠍ(25)ᑼ߳ઍ౉ߒߡ㧘ߎߩᑼߩฦ㗄ߪ ᰴᑼߩࠃ߁ߦ⴫ߐࠇࠆ㧚ߚߛߒ㧘◲නߩߚ߼ߦ㧘ฦ ⷐ⚛ߩ✚๺⸥ภߪ⋭⇛ߔࠆ㧚 1 1 1 2 3 1 1 1 1 1 1 2 3 _{1 1 1} ( ) ( | | ) [( ) ( ) ( ) ] ( ) ( ) ( L L t t e e e L _V L L t t t t t e L _V e e m e m m t t t t t e e e L e e d d t t t e dV J r d d d d d d  K K K K  K K K K K GU M M GU [ [ [ G U [ [ [ G U        ’ ˜  ˜        >    

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r N A N D A B N D A B N D A B d D N B A B S S B B S         _  M M M K K K M K _{2 3} 1 1 1 1 1 1 2 3 1 1 1 1 1 1 ) ( ) ] [( ) ( ) ( ) ( ) ] [( ) ( ) ( t t t m m m r e t t t t t m L m m e e m m e t t t t m m m m m m m r e t t t t e L d r e e r e t r m d d d d d d K K K  K K  K K [ [ [ G U [ [ [ G U                       

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d d t d S B B S S B d D N B A B S S B B S S B B S S B d d D N B A B S S B B S         ޓޓ K K M K K M K 2 3 ) ( t ) ] (28.1) m m r  r ed d d[ [ [ t t d d d S B K B S S B d bbbbbbbbbbb 1 1 1 2 3 1 1 1 1 1 1 2 2 3 1 1 1 1 1 1 2 2 3 1 1 1 1 1 1 1 1 1 1 _/ 2 1 [ / / ] 2 1 [ / / ] 2 [ / L t t L _V L e L e e e t t t t t e L e e e e e e e t t t t t m L m e e m e e e e t e L e e dV J d d d J D J d d d J D J d d d J  K   U G M M G U [ [ [ G U [ [ [ G U [ [ [ G U             ’ ˜’     

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t d A A S A A A S A A A d S A M M K M M M K M M M M 2 2 3 1 _{/ ] (28.2)} 2Dd e e e et t t J d d d[ [ [   M A AM 0 1 1 1 0 2 3 1 1 1 1 1 1 0 2 3 1 1 1 1 1 1 0 2 3 1 1 1 { ( )} ( ( ) )} [ ( [ ( L L t L _V L L _V r z t L t t t e L e d e t t t m L c m e d e a dV a r a z g z z rdrd dz a J g y y J d d d J D d d d J D d d d  K K K  K  U G U G T G U [ [ [ G U [ [ [ G U [ [ [ G          ˜    ˜      @     @ 

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r r B r B B d a B r B B d a K K K K 1 1 1 0 2 3 1 1 1 [ ( (28.3) t t t e

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_{  }UL d J D d  K e d e@ d d d[ [ [  d B r BK B d a   ߎߎߦ㧘a v ߢ޽ࠅ㧘࿾േടㅦᐲࡌࠢ࠻࡞ࠍ⴫ߔ㧚₀ (25)ᑼߩ╙ 4 㗄ߪኈེߩ㔌ᢔ♽ㆇേᣇ⒟ᑼߣߥࠆ㧚 0 [ { ( ) ( )} ] ( ) s E s A t e e e e h · 2 r dxd dt G U 3 T G    

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u u / v u d Md Kd f    V V  ޓޓ   (28.4) ߎࠇࠄߩᄌಽᑼࠍ㓸⸘ߔࠆߣ㧘ᰴᑼߩࠃ߁ߥ㔌ᢔ ♽ߩ႐ߩᣇ⒟ᑼࠍᓧࠆ㧚 1 2 1 2 1 12 1 12 2 12 2 12 e e t t e e t t t t e e d d d e e M I M M K K K ij ij 0 S S 0 K K K 0 f Ș Ș S S S 0 K K K 0 f d d S S S K K K 0 f d d 0 0 I 0 0 0 0 0 0           ­ ½ ­ ½ ª º ª º ­ ½ ° ° ° ° « »_{° °} « »_{° ° ° °}° ° « »_{® ¾} « »_{® ¾ ® ¾} « »_{° °} « »_{° ° ° °} « »_{° °} « »_{° ° ° °} ¬ ¼¯ ¿ ¬ ¼¯ ¿ ¯ ¿ (29)        _{(2)ᑼߩߺࠍ㔌ᢔൻߔࠆߣ㧘਄ᑼߩⴕ೉}S_g㧘߅ࠃ ߮ⴕ೉S_d㗄ߪ࠯ࡠߢ޽ࠆ㧚ߎࠇࠄߩ㗄ߪ♽ߦᷫ⴮ 㗄ࠍዉ౉ߒߚߣ߈ߦ㕖࠯ࡠߣߥࠆ㧚    Rq Kq f 0_{            (30)} ߎߎߦ㧘ᧂ⍮ߩࡌࠢ࠻࡞q { ,M K  e e d de e}ߢ޽ࠆ㧚  㧟႐ߩᣇ⒟ᑼߩ⸃ᴺ16,19)   ࿕᦭ᝄേᢙ  ⴕ೉KMߪ․⇣detKM 0ߢ޽ࠅ㧘ᰴᑼߩ᦭ᗧߢή ޿⸃ࠍ᦭ߔࠆ㧚 ^ ` 0 , 0 1, ,1 KMM  0 M       (31) ਄ᑼࠍ೑↪ߒߡ㧘ᰴߩᒻᑼߩㅒⴕ೉ࠍᓧࠆ㧚 1 0 1 2 0 0 e e t e K S S d 0 0 M M K M M O  ª º ­ ½ ­ ½ ª º  ® ¾ « » « »® ¾ ¯ ¿ ¬ ¼ ¬ ¼¯ ¿     (32)

 ߎࠇࠍ೑↪ߒߡ㧘ᰴᑼߩㆇേᣇ⒟ᑼࠍᓧࠆ㧚  1 2 1 12 0 1 0 2 0 12  §_{ª º ª º ª º ª º ° °·­ ½ ª º­ ½° °} ¨_{« » « » « » « »}¸_{® ¾ « »® ¾} ¨_{¬ ¼ «¬ ¼»«¬ »¼ ¬ ¼}¸_{° ° «¬ »¼¯ ¿° °} ¯ ¿ © ¹  t e  g e t t t e d e S S 0 0 K K K 0 0 S 0 ₀ d 0 M S 0 d K K   M M K K M (33)   ߎߩᣇ⒟ᑼࠍ㧘ኻ⒓ⴕ೉ࠍᜬߟ৻⥸࿕᦭୯໧㗴߳ ⺃ዉߒߡ㧘࿕᦭ᝄേᢙࠍᓧࠆߎߣ߇ߢ߈ࠆ㧚  ᔕ╵ߩ⸃ᨆᚻᴺ ᤨ㑆Ⓧಽߦ↪޿ࠆ⸃ᴺߩㆬᛯߪ㊀ⷐߥ್ᢿ੐㗄 ߩ߭ߣߟߢ޽ࠆ㧚ߎߩ໧㗴ߦㆡ↪ߒᤃߊ㧘቟ቯᕈ߇ ޽ࠆߣߒߡᐢߊᵹ૕໧㗴ߩ⸃ᨆߦ೑↪ߐࠇߡ޿ࠆ Crank㨯Nicolson(ࠢ࡜ࡦࠢ㨯࠾ࠦ࡞࠰ࡦ)ᴺ19)_ࠍណ↪ߔ ࠆ㧚_{n ࠬ࠹࠶ࡊߩ⻉㊂߇ᣢ⍮ߣߒߡ㧘ߎߩᣇᴺࠍ} (30)ᑼ߳ㆡ↪ߔࠆߣ㧘(n+1)ࠬ࠹࠶ࡊߩߟࠅ޽޿ᑼߪ ᰴᑼߩࠃ߁ߦ⴫⃻ߐࠇࠆ㧚 1 1 1 1 1 ( ) n n ( ) n n _'t n n n n n n   q q     R R K q K q f f 0 (34) ߎߩᑼߪ㧘ࠬ࠹࠶ࡊਛᄩߦ߅ߌࠆ⻉㊂_{(ᐔဋ୯)ࠍ} ↪޿ߡࠬ࠹࠶ࡊ㑆ߩㅦᐲࠍቯ߼ࠆߎߣࠍ⴫ߔ㧚ᵹ૕ ߣᒢᕈኈེߩᄢᄌᒻേ⊛ㅪᚑ໧㗴ߢ޽ࠅ㧘㕖✢ᒻᕈ ߇㜞޿ߣ੍᷹ߐࠇࠆߩߢ㧘ฦࠬ࠹࠶ࡊߦ߅޿ߡ㧘෼ ᧤⸘▚ࠍⴕ߁㧚_{(n+1)ࠬ࠹࠶ࡊߦ߅޿ߡ㧘࠾ࡘ࡯࠻} ࡦ㨯࡜ࡊ࠰ࡦᴺࠍ↪޿ߚ෼᧤Ṷ▚ࠍⴕ߁㧚_(n+1)ࠬ࠹ ࠶ࡊߩJ ࿁⋡ߩㄭૃ୯ࠍ 㧘ᰴ࿁ߩჇಽࠍ ߣ ߔࠆߣ㧘Ⴧಽᑼߪᒻᑼ⊛ߦᰴᑼߩࠃ߁ߦ⴫ߐࠇࠆ㧚 1 1 1 1 1 1 1 1 1 1 1 1 1 ( ) ( ) 1 [ )] {( ) ( )} n n n n n n n n n n t t t n n n n n n n n n n t t t J J J J ' ' ' '               _w  _{ }w _w w w w        R R R q q _K K q f _q q q q q q R R K q K q f f bbbbbb ♽ߩᔕ╵ߪ㧘_{(35)ᑼࠍ↪޿ߡ▚ቯߐࠇࠆ㧚} 㧠.ᢙ୯⸃ᨆ  ⸃ᨆࡕ࠺࡞ߣᔕ╵⸃ᨆߩၮᧄቯᢙ  ߎߎߢ೑↪ߔࠆ౞╴࠲ࡦࠢߩᢙ୯⸃ᨆࡕ࠺࡞ࠍ␜ ߔ㧚࠲ࡦࠢߪ_{2 ゲኻ⒓ߩᒻ⁁ࠍ᦭ߔࠆ౞╴ࠪࠚ࡞ߢ} ᭴ᚑߐࠇࠆ㧚ࠪࠚ࡞⣉ㇱߪࡇࡦᡰᜬߣߔࠆ㧚౞╴ࠪ ࠚ࡞㧘߅ࠃ߮ᵹ૕ߪ๟ᣇะߦ_{16 ୘ߩⷐ⚛ࡔ࠶ࠪࡘ} ࠍ᦭ߔࠆ㧚౞╴ࠪࠚ࡞ߪᲣ✢ᣇะߦ_{4 ୘ߩⷐ⚛ࡔ࠶} ࠪࡘࠍ᦭ߔࠆ㧚ᵹ૕ߪඨᓘᣇะߦ_{5 ጀ㧘਄ਅߦ 6 ጀ} ߩࡔ࠶ࠪࡘࠍᜬߟ㧚㐽ᦛ㕙_{(close)ߩኈེߢ޽ࠅ㧘ᵹ} ૕ߩඨᓘᣇะߩⷐ⚛㧘๟ᣇะߩⷐ⚛ᢙ㧘߅ࠃ߮ᵹ૕ ߩ਄ਅߩⷐ⚛ᢙߣኈེߩ਄ਅߩⷐ⚛ᢙߩᢙሼࠍਗߴ ߡ㧘⸥ภC5 16 6 4u u u ࠍ↪޿ߡ⴫ߔ㧚 ࠲ࡦࠢߩඨᓘr_sߪ25m㧘ኈེ㜞ߐH ߪ 30m㧘ᵹ ૕㜞ߐߪ_{21.6m ߣߔࠆ㧚ኈེෘߐh ߪ 40mm㧘߅ࠃ} ߮_{55mm ࠍ↪޿ࠆ㧚ኈེߩⷐ⚛ಽഀߣᵹ૕ߩ⥄↱⴫} 㕙ߩⷐ⚛ࡔ࠶ࠪࡘࠍࠕࠗ࠰ࡔ⴫␜ߒߡ㧘_{Fig.2 ߦ␜} ߔ㧚_{(35)ᑼߩ⁛┙⥄↱ᐲߪ 1384 ୘(ㅦᐲࡐ࠹ࡦࠪࡖ} ࡞_{623 ୘㧘ᵄ㜞 89 ୘㧘ࠪࠚ࡞ 672 ୘)ߢ޽ࠆ㧚} ᷫ⴮ߪಽᢙ⺞ᵄᝄേߩ↢⿠ߦᄢ߈ߥᓇ㗀ࠍ᦭ߔࠆ ߎߣ߇⍮ࠄࠇߡ޿ࠆ㧚ߎߎߢߪ㧘ᰴߩᷫ⴮16,19)ࠍ઒ ቯߒߚ㧚⥃⇇ᷫ⴮ᲧߪᝄേᢙZK 0.82rad/s㧘߅ࠃ ߮Z _{c 7.0rad/s}ߦኻߒߡ㧘hc hK 0.0075ࠍਈ߃ߚ㧚 ߹ߚ㧘ᤨೞᱧᔕ╵ߪᤨ㑆ೞߺӠ_{t=0.01 ⑽ߣߒ㧘ᤨ} ೞ_{100 ⑽߹ߢߩᔕ╵ࠍ⸃ᨆߒߚ㧚(35)ᑼߦ߅ߌࠆ࠾} ࡘ࡯࠻ࡦ㨯࡜ࡊ࠰ࡦᴺߩ෼᧤Ṷ▚ߪ_{4 ࿁ߣߒߚ㧚}  Fig.2C5 16 6 4u u u Model  ᔕ╵࠺࡯࠲ߩᢛℂ 4.2.1 ࠲ࡦࠢᔕ╵ߩ⸘᷹૏⟎ߣ๟ᣇะዷ㐿ᰴᢙ  ᵹ૕ߩᵄ㜞㧘߅ࠃ߮ኈེߩᄌ૏㧘วᔕജߩ᷹ቯ૏ ⟎ߣ⴫␜ᣇᴺࠍ␜ߔ㧚ኈེߩᏅಽᄌ૏㧘߅ࠃ߮วᔕ ജߪ⥄↱⴫㕙߆ࠄ_{-6m ૏⟎㧔࠲ࡦࠢᐩ߆ࠄ 15.6m㧕} ߩᔕ╵୯ࠍ↪޿ߚ㧚ኈེߩ๟ᣇะᴺ✢ᣇะߩᏅಽᄌ ૏㧘Ꮕಽวᔕജ㧘߅ࠃ߮๟ᣇะᵄ㜞ߪ᦭㒢ࡈ࡯࡝ࠛ ⚖ᢙ߳ዷ㐿ߒߚࡈ࡯࡝ࠛଥᢙ㧔ࠬࡍࠢ࠻࡞ߣ⸥ㅀߔ ࠆ㧕ࠍ↪޿ߚ⴫␜߽೑↪ߔࠆ㧚ኈེߩᔕജ߽ߎߩ࡟ ࡌ࡞ߩวᔕജ_{NT,߅ࠃ߮ MT}ࠍ↪޿ߡ㧘࿑␜ߔࠆ㧚 1 n+J q 'qn+1 (35) b

ࡈ࡯࡝ࠛ⚖ᢙߩ๟ᣇะዷ㐿ᰴᢙࠍ_{n ߣ⴫␜ߒ㧘ടജ} ᣇะߦ㑐ߒߡኻ⒓ߥᵄ㜞㧘߅ࠃ߮ᄌ૏ࡕ࡯࠼ࠍ Cosine ᚑಽ㧘ㅒኻ⒓ߣߥࠆࡕ࡯࠼ࠍ Sine ᚑಽߣ⸥ ㅀߔࠆ㧚ߎࠇࠄߩᚑಽߪߘࠇߙࠇ๟ᣇะዷ㐿ᰴᢙߩ ᚑಽࠍᜬߟߩߢ㧘_{Cosine ᚑಽ㧔C0,C1,},C8㧕㧘߅ࠃ ߮_{Sine ᚑಽ(S1,S2,},S7㧕ߩ⴫␜ࠍ೑↪ߔࠆ㧚 4.2.2 㕖✢ᒻᝄേᔕ╵ߩᝄേࡕ࡯࠼ߩ․ቯ ಽጘߔࠆ㕖✢ᒻᝄേᔕ╵ࠍ᭴ᚑߔࠆᝄേࡕ࡯࠼ࠍ ․ቯߔࠆߚ߼ߦ㧘࿕᦭ᝄേᢙZ_{n k.} ߦኻᔕߔࠆ࿕᦭ᝄ േࡕ࡯࠼ c nk I 㧘߅ࠃ߮ s nk I ࠍ↪޿ߡ㧘ᔕ╵ࠍᰴᑼߩࠃ ߁ߦಽᨆߔࠆ㧚છᗧᤨೞߩᔕ╵q { , , }ij Ș d_{e e e} ߣߔ ࠆߣ߈㧘ߎߩ߁ߜࡌࠢ࠻࡞ij ࠍ⋭⇛ߒ,ᱷࠆࡌࠢ࠻_e ࡞q { , }Ș d_{e e} ࠍᝄേࡕ࡯࠼ߩᒛࠆⓨ㑆ߦዷ㐿ߔࠆ㧚 c c s s 0 1

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      (36) ߎߎߢߪ㧘๟ᣇะߦࠪࠚ࡞㧘߅ࠃ߮ᵹ૕ࠍ_{16 ╬ಽ} ߒߡ޿ࠆߩߢ㧘_{n ߪ 0 ߆ࠄ 8 ߹ߢࠍߣࠅ㧘n=0,8 ߦ} ኻߔࠆ  ߪሽ࿷ߒߥ޿㧚 ᤨೞᱧ_{89.77 ⑽߆ࠄ 100 ⑽߹ߢߩ㧘ᤨ㑆㑆㓒 0.01} ⑽ߩᔕ╵_{1024 ୘ࠍ਄⸥ᝄേࡕ࡯࠼ߦಽ⸃ߔࠆ㧚ߘ} ߩᓟ㧘ฦᝄേࡕ࡯࠼ߩᔕ╵ࠍ᦭㒢ࡈ࡯࡝ࠛ⚖ᢙߦዷ 㐿ߒߡ㧘ᔕ╵ࠬࡍࠢ࠻࡞ࠍ⸃ᨆߒ㧘ಽጘߔࠆᔕ╵ߩ ᝄേࡕ࡯࠼ࠍ․ቯߔࠆ㧚ᧄ⺰ᢥߢߪ㧘ᦨዊ࿕᦭ᝄേ ᢙ߆ࠄ_{144 ⇟⋡ߩ࿕᦭ᝄേᢙߦኻᔕߔࠆ࿕᦭ᝄേࡕ} ࡯࠼ࠍ೑↪ߒߡ㧘ᔕ╵ߩᝄേࡕ࡯࠼ಽᨆࠍⴕ߁㧚 4.2.3 ᔕ╵ᚑಽߩࠬࡍࠢ࠻࡞ಽᨆ  ᧄ⺰ᢥߦ↪޿ߚᵄ㜞㧘߅ࠃ߮ኈེߩᏅಽᄌ૏㨯ว ᔕജ╬ߩᔕ╵ࠬࡍࠢ࠻࡞ࠍ␜ߔ㧚ᤨ㑆㑆㓒_{0.01 ⑽} ࠍ↪޿ߡ⸃ᨆߐࠇߚᤨೞᱧ_{89.77 ⑽߆ࠄ 100 ⑽߹ߢ} ߩ_{1024 ୘ߩ࠺࡯࠲ࠍ᦭㒢ࡈ࡯࡝ࠛ⚖ᢙߦዷ㐿ߒ㧘} ᓧࠄࠇߚࡈ࡯࡝ࠛଥᢙࠍᔕ╵ࠬࡍࠢ࠻࡞ߣߒߚ㧚 ࿕᦭ᝄേᢙߩᢙ୯⸃ᨆ ኈེߪ㍑᧼㧘ᵹ૕ߪ᳓ࠍࡕ࠺࡞ߣߒ㧘᧚ᢱߩᯏ᪾ ⊛ᕈ⾰ࠍ_{Table1 ߦ␜ߔ㧚4.1 ▵ߦ␜ߒߚ⸃ᨆࡕ࠺࡞} C5 16 6 4u u u ߩኈེࠪࠚ࡞ෘߐ_{h ࠍ 40mm㧘߅ࠃ߮} 55mm ߣߒߚ♽ߩ࿕᦭ᝄേᢙࠍߘࠇߙࠇ Table2,߅ࠃ ߮_{Table3 ߦ␜ߔ㧚ࠪࠚ࡞᧼ෘ߇ᄌൻߒߡ߽ᝄേᢙ} ߇৻ቯߩ୯ࠍ␜ߔ࿕᦭ᝄേᢙߪࠬࡠ࠶ࠪࡦࠣᝄേ㧘 ޽ࠆ޿ߪࠪࠚ࡞㕙ౝᄌᒻߦኻᔕߔࠆᝄേࡕ࡯࠼ߦኻ ᔕߔࠆ㧚᧼ෘߩᓇ㗀ࠍฃߌࠆ࿕᦭ᝄേᢙߦኻᔕߔࠆ ᝄേࡕ࡯࠼ߪࠪࠚ࡞㕙ᄖᄌᒻ߇Ყセ⊛ᄢ߈޿ߣ್ᢿ ߢ߈ࠆ㧚ߎࠇࠄߩᝄേࡕ࡯࠼ߩ᜼േ߇ᵹ૕ߣኈེߩ 㕖✢ᒻᝄേᔕ╵ߦᄢ߈ߥᓇ㗀ࠍ෸߷ߔ㧚 Table2 ߦ␜ߒߚ๟ᣇะዷ㐿ᰴᢙ n=5 ߦኻᔕߔࠆ 6㧘 8㧘9 ⇟⋡ߩ࿕᦭ᝄേᢙZ Z ,߅ࠃ߮5.6, 5.7 Z ߦኻᔕߔ5.8 ࠆᝄേࡕ࡯࠼ߩࠕࠗ࠰ࡔ࿑ߣᐔ㕙ᛩᓇ࿑ࠍ_Fig.3.1 ߦ␜ߔ㧚 _{n=6 ߦኻᔕߔࠆ 5,6,߅ࠃ߮ 8 ⇟⋡ߩ࿕᦭ᝄ} േᢙZ Z ,߅ࠃ߮_6.5, _6.6 Z ߦኻᔕߔࠆᝄേࡕ࡯࠼ߩࠕ   _6.8 㩷㩷㩷㩷㩷㩷㩷㩷㩷㩷㩷㪫㪸㪹㫃㪼㪅㪈䇭㪤㪸㫋㪼㫉㫀㪸㫃㩷㪧㫉㫆㫇㪼㫉㫋㫐 )OXLG 0DVV 'HQVLW\ 'HQVLW\0DVV 0RGXOXV<RXQJ 3RLVVRQ5DWLR NJP_NJP_{*3D } 6KHOO6WHHO

Table.2 Natural Frequency of a Cylindrical Tank(h =40mm)

n 1 2 3 4 5 6 7 8 9 0 1.08 1.33 1.79 2.28 2.77 3.08 32.20 33.12 35.01 1 0.82 1.49 1.99 2.50 2.87 15.12 17.55 25.14 31.56 0.82 1.49 1.99 2.50 2.87 15.12 17.55 25.14 31.56 2 1.11 1.70 2.20 2.71 2.99 9.98 17.49 19.11 27.85 1.11 1.70 2.20 2.71 2.99 9.98 17.49 19.11 27.85 3 1.33 1.89 2.38 2.84 3.42 8.06 16.51 23.63 30.41 1.33 1.89 2.38 2.84 3.42 8.06 16.51 23.63 30.41 4 1.53 2.08 2.57 2.95 3.93 7.17 14.98 17.41 18.37 1.53 2.08 2.57 2.95 4.23 7.17 14.98 18.37 5 1.73 2.28 2.77 3.12 4.83 6.73 12.72 13.49 17.43 1.73 2.28 2.77 3.12 4.83 6.73 12.72 13.49 17.43 6 1.91 2.47 2.94 3.38 5.59 6.47 9.23 11.66 15.70 1.91 2.47 2.94 3.38 5.59 6.47 9.23 11.66 15.70 7 2.06 2.62 3.08 3.65 6.22 6.37 7.53 9.48 12.29 2.06 2.62 3.08 3.65 6.22 6.37 7.53 9.48 12.29 8 2.12 2.69 3.14 3.77 6.51 6.66 6.99 7.90 9.55

Table.3 Natural Frequency of a Cylindrical Tank(h =55mm)

n 1 2 3 4 5 6 7 8 9 0 1.08 1.33 1.79 2.28 2.77 3.08 44.28 48.13 52.75 1 0.82 1.49 1.99 2.50 2.87 17.55 22.25 35.39 43.22 0.82 1.49 1.99 2.50 2.87 17.55 22.25 35.39 2 1.11 1.70 2.20 2.71 2.99 14.89 17.49 22.25 37.84 1.11 1.70 2.20 2.71 2.99 14.89 17.49 22.25 37.84 3 1.33 1.89 2.38 2.85 3.42 11.92 22.35 30.74 33.58 1.33 1.89 2.38 2.85 3.42 11.92 22.35 30.74 33.58 4 1.53 2.08 2.57 2.95 3.93 10.36 17.41 19.32 21.71 1.53 2.08 2.58 2.96 4.23 10.36 19.32 21.71 5 1.73 2.28 2.78 3.13 4.83 9.26 14.67 17.43 18.22 1.73 2.28 2.78 3.13 4.83 9.26 14.67 17.43 18.22 6 1.92 2.48 2.95 3.39 5.59 8.26 11.86 15.58 19.63 1.92 2.48 2.95 3.39 5.59 8.26 11.86 15.58 19.63 7 2.07 2.64 3.09 3.65 6.23 8.02 10.41 12.67 15.75 2.07 2.64 3.09 3.65 6.23 8.02 10.41 12.67 15.75 8 2.13 2.70 3.16 3.77 6.53 8.55 9.61 10.86 13.13 s nk

I

ع㧘߅ࠃ߮⸥ภغࠍઃߒߡ㧘Figs.16, 26, 36,߅ࠃ߮ Fig.46 ߦ␜ߒߚ㧚 5.1.4 ߩࡕ࠺࡞ߩኈེߦធߔࠆ⥄↱⴫㕙਄ߦ߅ߌ ࠆᵄ㜞ߩCosine ᚑಽ㧘߅ࠃ߮ Sine ᚑಽߩᔕ╵ࠬࡍ ࠢ࠻࡞ࠍ Fig.53㧘߅ࠃ߮ Fig.54 ߦ␜ߔ㧚ಽᢙ⺞ᵄᝄ േߩᝄേᢙߣථ⿧ࠬࡍࠢ࠻࡞ࠍߣࠆ๟ᣇะዷ㐿ᰴᢙ ࡕ࡯࠼n ߪ Fig.34,߅ࠃ߮ Fig.35 ߣห৻ߢ޽ࠆ㧚ࠬ ࡠ࠶ࠪࡦࠣᝄേߩࡕ࡯࠼ߩCosine ᚑಽ㧘߅ࠃ߮ Sine ᚑಽߩᔕ╵ߪߘࠇߙࠇᝄേᢙ 0.61rad/s ߢ޽ࠆ㧚 5.1.5 ߩࠤ࡯ࠬߢ߽ਥⷐߥ 4 ୘ߩࠬࡍࠢ࠻࡞ࠍ᦭ߔ ࠆᔕ╵ࠍ␜ߔ㧚ߎࠇࠄߩࠬࡍࠢ࠻࡞ߩᏅ߽0.61rad/s ߣߥࠆ㧚 5.2.4 ๟ᣇะዷ㐿ᰴᢙߣᔕ╵ࠬࡍࠢ࠻࡞  න৻ߩࠬࡍࠢ࠻࡞ࠍ᦭ߔࠆಽᢙ⺞ᵄᝄേ↢⿠ߔࠆ 5.1.4㧘߅ࠃ߮ 5.1.5 ߩࠤ࡯ࠬߦ߅޿ߡ㧘๟ᣇะዷ㐿 ᰴᢙn ߇஧ᢙ㧘߅ࠃ߮ᄸᢙࠍ᦭ߔࠆᝄേࡕ࡯࠼ߩಽ ᢙ⺞ᵄᝄേߩࠬࡍࠢ࠻࡞ߪหߓߢ޽ࠆ㧚 ߒ߆ߒߥ߇ࠄ㧘৻⥸⊛ߦ๟ᣇะዷ㐿ᰴᢙ߇஧ᢙ㧘 ޽ࠆ޿ߪᄸᢙߦኻᔕߒߡ㧘ߘࠇߙࠇ⇣ߥࠆᝄേᢙࠍ ᜬߟಽᢙ⺞ᵄᝄേᔕ╵߇↢⿠ߒᓧࠆ㧚਄⸥ߩ2 ࠤ࡯ ࠬߢߪCosine ᚑಽߦ๟ᣇะዷ㐿ᰴᢙ߇ᄸᢙߩᝄേ ࡕ࡯࠼߇ಽጘߒߚᤨ㧘Sine ᚑಽߪ๟ᣇะዷ㐿ᰴᢙ߇ ஧ᢙߩᝄേࡕ࡯࠼߇ಽጘߒߡ޿ࠆ㧚5.1.2 ߩࠤ࡯ࠬ ߪ๟ᣇะዷ㐿ᰴᢙ߇஧ᢙ,߅ࠃ߮ᄸᢙߩᝄേࡕ࡯࠼ ߇߶߷╬ߒ޿ᝄേᢙߩࠬࡍࠢ࠻࡞ࠍᜬߞߚߣℂ⸃ߔ ࠆߎߣ߇ߢ߈ࠆ㧚5.1.3 ߪ Sine ᚑಽߩᔕ╵߇߶߷࠯ ࡠߢ޽ࠆߣߒߚࠤ࡯ࠬߣߥࠆ㧚 ⶄᢙߩࠬࡍࠢ࠻࡞ࠍ᦭ߔࠆಽᢙ⺞ᵄᝄേᔕ╵ࠍ 5.1.4㧘߅ࠃ߮ 5.1.5 ߦ␜ߒߚ㧚 㧢. ⚿⺰ 1. ౞╴࠲ࡦࠢߦ߅ߌࠆࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢᕈ౞ ╴ࠪࠚ࡞ኈེߩᄢᄌᒻ႐ߦ߅ߌࠆേ⊛ㅪᚑ໧㗴 ߩ᳢㑐ᢙ߳᦭㒢ⷐ⚛ᴺࠍ↪޿ࠆ⋥ធᴺࠍㆡ↪ߒ ߡ㧘ቯᑼൻߒ㧘⸃ᨆߔࠆᣇᴺࠍ␜ߒߚ㧚 2. ᵹ૕߇৻᭽⤘ᒛߔࠆ᦭ᗧߢή޿⸃ࠍឃ㒰ߔࠆᚻ ᴺࠍ␜ߒ㧘ㅪᚑ♽ߩ࿕᦭ᝄേᢙࠍ⸃ᨆߒ㧘ߎߩ ♽ߩ࿕᦭ᝄേࡕ࡯࠼ࠍ࿑␜ߒߚ㧚 3. ᱜᒏᵄߩ࿾േࠍฃߌࠆ౞╴࠲ࡦࠢߩᔕ╵ࠍ⸃ᨆ ߒ㧘ᄖജᝄേᢙߣᄖജߩᄢ߈ߐࠍࡄ࡜ࡔ࡯࠲ߣ ߔࠆಽᢙ⺞ᵄᝄേ߇ಽጘߔࠆਇ቟ቯ㗔ၞ߇޽ࠆ ߎߣࠍ␜ߒߚ㧚 4. ਄⸥ਇ቟ቯ㗔ၞߦ߅޿ߡ㧘ಽጘߔࠆಽᢙ⺞ᵄᝄ േ߇න৻ߩᝄേᢙࠍ᦭ߔࠆὐ߇޽ࠆߎߣࠍ␜ߒ㧘 ᔕ╵ࠬࡍࠢ࠻ߩಽᨆ㧘ಽᢙ⺞ᵄᝄേᔕ╵߇⊒↢ ߒߚᝄേࡕ࡯࠼ߩ․ቯࠍⴕߞߚ㧚1/2 ಽᢙ⺞ᵄᝄ േ㧘޽ࠆ޿ߪ1/3 ಽᢙ⺞ᵄᝄേ߇↢⿠ߔࠆὐ߇ ޽ࠆߎߣࠍ␜ߒߚ㧚 5. න৻ߩᝄേᢙࠍ᦭ߔࠆಽᢙ⺞ᵄᝄേ߇⊒↢ߔࠆ ὐߩㄭறߦ߅޿ߡ㧘ⶄᢙߩᝄേᢙࠍᜬߟಽᢙ⺞ ᵄᝄേ߇↢⿠ߔࠆ㧚ߎߩࠃ߁ߥࠤ࡯ࠬߩᔕ╵ࠍ ␜ߒߚ㧚߹ߚ㧘ኈེߩᏅಽᄌ૏㧘߅ࠃ߮Ꮕಽว ജߩᔕ╵߽␜ߒ㧘ಽᢙ⺞ᵄᝄേᚑಽ߿㜞⺞ᵄᝄ േᚑಽ߇⊒↢ߔࠆߎߣࠍ␜ߒߚ㧚 6. ᱜᒏᵄߩ࿾േ߇૞↪ߔࠆᤨ㧘Ყセ⊛ዊߐߥᄖജ ߩ߽ߣߢ⊒↢ߔࠆಽᢙ⺞ᵄᝄേᔕ╵ߦὶὐࠍว ࠊߖߡᬌ⸛ߒߡ߈ߚ㧚ߎࠇࠄߩㆊ⒟ߦ߅޿ߡ㧘 ឭ᩺ߒߚᚻᴺ߇࠲ࡦࠢߦ߅ߌࠆᵹ૕ߣኈེߩᄢ ᄌᒻㅪᚑ႐ߩਇ቟ቯߥ᜼േࠍචಽߥ♖ᐲࠍᜬߞ ߡ⸃ᨆߢ߈ࠆᚻᴺߢ޽ࠆߎߣࠍታ⸽ߒߚ㧚ᄖജ ߇ᄢ߈ߥ㗔ၞߢ↢⿠ߔࠆಽᢙ⺞ᵄᝄേߩ᜼േ㧘 ࿾㔡ߦࠃࠆਇⷙೣߥ࿾േ߇૞↪ߒߚᤨߩᔕ╵߿ ቟ోᕈࠍᬌ⸛ߔࠆߎߣߪᰴߩ⺖㗴ߢ޽ࠆ㧚 ⻢ㄉ  ⥄૞ߩ᳓ᐔᝄേบߦ᳓ߩ౉ߞߚ౞╴࠲ࡦࠢࠍ⸳⟎ ߒ㧘8 ࡆ࠶࠻ߩࡄ࠰ࠦࡦߦขઃߌߚ AD ᄌ឵ࡏ࡯࠼ ࠍ೙ᓮߒߚ᷹ቯⵝ⟎ࠍ↪޿ߡ㧘ᔕ╵ࠍ᷹ቯߔࠆታ㛎 ࠍ1984 ᐕߦᆎ߼߹ߒߚ㧚ߘࠇએ᧪ᄙߊߩቇ↢㧘߅

ࠃ߮㒮↢߇ߎߩᝄേታ㛎ࠍ⛮⛯ߒ㧘⡊ࠍߟࠎߑߊ㖸 ߣ౒ߦᕆỗߦಽጘߔࠆಽᢙ⺞ᵄᝄേᔕ╵ߩ․ᕈࠍᛠ ីߔࠆ⎇ⓥߦᓥ੐ߒ߹ߒߚ㧚ᓐࠄߩ₂り⊛ߥദജߦ ⴲᔃ߆ࠄᗵ⻢ߒ߹ߔ㧚ታ㛎ߢⷰኤߐࠇߚߎߩᝄേ⃻ ⽎ࠍℂ⺰⊛ߦౣ⃻ߔࠆ߹ߢ㧘_{29 ᐕࠍ⾌߿ߒ߹ߒߚ㧚} ᧄ⎇ⓥߪ_{JSPS ⑼⎇⾌ 23560677 ߩഥᚑࠍฃߌߚ߽} ߩߢߔ㧚 ઃ㍳_{1 ౞╴ࠪࠚ࡞ߩ᦭㒢ᄌᒻ߭ߕߺ⴫⃻}  ᦭㒢ᄌᒻߩ౞╴ࠪࠚ࡞ߦ߅޿ߡ೰૕ᄌ૏ࠍ㒰෰ߒ ߚ߭ߕߺ⴫⃻ࠍ⷗ࠆߎߣߪ⒘ߢ޽ࠅ㧘◲නߦ⺃ዉߔ ࠆ㧚౞╴ࠪࠚ࡞ߩඨᓘࠍr_sߣߔࠆߣ㧘_Novozhilov20) ߩ↪޿ߚ⴫⸥ߪᰴᑼߩࠃ߁ߦ⴫ߐࠇࠆ㧚 11 , 21 , 12 , 22 , 13 23 , , ˆ ,ˆ / ,ˆ ,ˆ ( ) / ,ˆ , ˆ , , ( ) / x s x s x x x s e u e u r e v e v w r e e w w v r T T T T T E E E E       (A1) ᐳᮡ♽ߩࡄ࡜ࡔ࡯࠲ߪᲣ✢ᣇะ㧘߅ࠃ߮๟ᣇะߪ ߘࠇߙࠇx㧘߅ࠃ߮Tߢ޽ࠆ㧚㧔_{A1㧕ᑼࠍ೑↪ߒߡ} 㕙ౝ߭ߕߺߪᰴᑼߩࠃ߁ߦ⴫ߐࠇ㧘ㄭૃߐࠇࠆ㧚 2 2 2 2 11 11 12 13 11 13 2 2 2 2 2 2 22 21 22 23 22 22 23 22 23 12 21 11 21 22 12 13 23 12 21 13 23 ˆ (ˆ ˆ ˆ ) / 2 ˆ ˆ / 2 ˆ (ˆ ˆ ˆ ) / 2 ˆ (ˆ ˆ ) / 2 ˆ ˆ / 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x e e e e e e e e e e e e e e e e e e e e e e e e e e e T H H J    |     |   |      |   (A2) ㄭૃᑼߩᦨᓟߩ⴫⃻߇ᧄ⺰ߩ߭ߕߺ⴫⃻ߢ޽ࠆ㧚 ᦛ₸㧘߅ࠃ߮ᝦ₸ߦ㑐ߒߡߪ㧘_{Sanders ℂ⺰ߦၮ} ߠ޿ߡ㧘೰૕ㆇേ߇ᱡࠍ↢ߓߐߖߥ޿ᰴᑼ21)ࠍណ ↪ߒߚ㧚 ,, , / , , , / ( , ,) / (2 ) x x x T T T rs Tx xT rs vx uT rs N E N E  F E E   _(A3) ߎࠇࠄࠍ↪޿ߡ㧘߭ߕߺࠛࡀ࡞ࠡ㑐ᢙߪᰴᑼߩࠃ߁ ߦ⴫ߐࠇࠆ㧚 2 2 2 2 3 2 2 2 2 ( ) / (1 ){( (1 ) / 2) / 2} / (1 ){( (1 ) / 2) / 2} x x x x Eh Eh T T T T 3 Q H H QH H Q J Q N N QN N Q F            u (A4) ߎߎߦ㧘E:ࡗࡦࠣ₸_,Q:ࡐࠕ࠰ࡦᲧ_,h:ࠪࠚ࡞ෘ㧚 ઃ㍳_{2 ౞╴ࠪࠚ࡞ߩᄌ૏㑐ᢙ}  ᧄ⺰ᢥߦ߅޿ߡ೑↪ߒߚ౞╴ࠪࠚ࡞ⷐ⚛ࠍ᣿␜ߔ ࠆ㧚ߎߩᄌ૏㑐ᢙ17,21,22)ߪᰴᑼߩࠃ߁ߦ⴫ߐࠇࠆ㧚 3 2 3 2 3 3 7 8 11 19 20 17 19 2 2 2 2 11 19 20 16 8 17 2 2 3 2 3 18 19 15 2 2 2 9 16 10 17 18 12 19 3 13 (3 / 4 - ) - / 2- / 6 ( / 4- ) / 6 / 2 / 2 / 6 ( ) ( - ) - - / 2-- / 6-u a x a x a R a R a R a R a R v a a R a R x a R a x R a R x a R a R x a x w a a R R a a R Rx a R a x a x R a x a I I I I I I I I I I               2 3 14xI/ 2-a x15 I/ 6 (A5) ߎߎߦ㧘dI dT ,R:ࠪࠚ࡞ඨᓘߢ޽ࠆ㧚ߎࠇࠄߩ ᄌ૏㑐ᢙߪᄌ૏u,v,w ߩㆡว᧦ઙࠍቢోߦḩߚߒߡ ޿ࠆ㧚ⷐ⚛ߩฦ▵ὐߦ߅ߌࠆ▵ὐᄌ૏ࡌࠢ࠻࡞ߪ (10)ᑼߩᧂቯଥᢙ 14 ୘㧘߅ࠃ߮೰૕ᄌᒻࠍ⴫ߔ 6 ୘ࠍട߃ߚ⸘_{20 ୘ߩࡄ࡜ࡔ࡯࠲ࠍᜬߟ㧚} (A5)ᑼߩᄌ૏㑐ᢙߪ㧘ਅ✢ઃ߈㗄ࠍήⷞߔࠆߣ㧘 Sabir ⷐ⚛22)_{ߣߥࠆ㧚Sabir ⷐ⚛ߪ㧘㜞♖ᐲߩ౞╴ࠪ} ࠚ࡞ⷐ⚛ߣ⹏ଔߐࠇߡ޿ࠆ㧚_{Sabir ⷐ⚛ߩᄌ૏㑐ᢙ} ߪ㧘ㄝߩᴺ✢ᣇะߩ࿁ォࠍᰴᑼߩࠃ߁ߦਈ߃ࠆ㧚 7 8 2 3 0 1 2 - 15 , ( , ) / /(6 ) x x a a a u w v R C C x C x x R R I I E I E     (A6)  ߎߎߦ㧘C C C₀, ,_{1 2}ߪ㨤ࠍ฽߹ߥ޿㗄ߢ޽ࠆ㧚 ▵ὐᄌ૏ߪᄌ૏ߣ࿁ォࠍ↪޿ߡ޿ࠆߩߢ㧘ᴺ✢ᣇ ะߩ࿁ォߪ_{2 ᰴએਅߩઍᢙ㑐ᢙߢ⴫ߐࠇࠆߎߣ߇ᢙ} ℂ⊛ߦㆡಾߢ޽ࠆ㧚 (A5)ᑼߩਅ✢ࠍઃߒߚ㗄ࠍዉ౉17)_{ߔࠆߣ㧘(A6)} ᑼߩ࿁ォE_Iߦ฽߹ࠇࠆ㨤ߩ3 ᰴ㗄ߪᶖṌߔࠆ㧚ߎ ߩᠲ૞ߪ㧘_{Sabir ⷐ⚛ߦ‛ℂ⊛ߥᄌᒻ᜔᧤ࠍട߃ࠆ} ߩߢ㧘ⷐ⚛೰ᕈࠍ㜞߼ࠆലᨐࠍᜬߟ㧚  ෳ⠨ᢥ₂

1) Luke,J.C. ,A variational principle for a fluid with a free surface,J.Fluid Mech,vol.27,part.2,pp.395-397 (1967). 2)Clough,R.W.,Niwa,A.,Clough.D.P.:Experimental

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3) Haroun,M.A.,Housner,G.A.Dynamic Characteristics of Liquid Storage Tanks: Complications in Free Vibration Analysis of Tanks,Proc.ASCE,Vol.108, No.EM5, pp.783-818 (1982).

4) ၳ⋥ੱ,⼱⾗ା,↰ਛᒎኼ㓶,ᶧ૕ߩ౉ߞߚ౞╴ࠪࠚ ࡞ߩേ⊛⸃ᨆ_{,ᣣᧄᑪ▽ቇળ⺰ᢥႎ๔㓸,╙ 282} ภ_{,pp.83-94 (1979).} 5) ᧻੗ᔀ຦,ᶋደᩮᑼ౞╴ᵹ૕⾂ᮏߩ࿾㔡ᤨߩࠬࡠ ࠶ࠪࡦࠣᔕ╵ߩ⸃ᨆ⸃_{,ᣣᧄᑪ▽ቇળ᭴ㅧ♽⺰ᢥ} 㓸╙_{594 ภ, pp.167-173 (2005).} 6) ౝᶏ㓷ᒾ,⍹↰๺㓶,ᣣ⹣㓷ਯ,㕖✢ᒻࠬࡠ࠶ࠪࡦࠣ ߦࠃࠆ⍹ᴤ࠲ࡦࠢᶋደᩮߩᝄേߦ㑐ߔࠆ⎇ⓥ_, IHI ᛛႎ,Vol.51,pp.55-62,No.1(2011). 7) ⊝Ꮉᵗ৻,᦭㒢ᄌᒻ႐ߢߩࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢ ᕈ૕ኈེߩ⋧੕૞↪ࠍᡰ㈩ߔࠆ᳢㑐ᢙ_{,ᣣᧄᑪ▽} ቇળ᭴ㅧ♽⺰ᢥႎ๔㓸╙_{362 ภ,pp.105-115} (1986).

8) Y.Minakawa, Lagrangian Functions of the Interactive Behavior Between Potential Fluid and Elastic Containers in Fields of Finite Deformations,Shells Membranes and Space Frames, Proceedings IASS Symposium,Osaka,Vol.1 ,pp.73- 80 (1986). 9) Y.Minakawa,Nonlinear Oscillation Analysis of

Interaction Behaviors Between the Potential Fluid and Tanks of Shell of Revolution in Finite Deformations,ᣣᧄᑪ▽ቇળ᭴ㅧ♽⺰ᢥႎ๔㓸 No.435 ,pp.91-107, (1992). 10) ጊᧄᙗม,⊝Ꮉᵗ৻,ቢోᵹ૕ߦ߅ߌࠆ㕖✢ᒻࠬ ࡠ࠶ࠪࡦࠣߩ᦭㒢ⷐ⚛⸃ᨆᴺ_{,ᣣᧄᑪ▽ቇળ᭴} ㅧ♽⺰ᢥ㓸 _{No.609,pp.89-96 (2006).} 11) ⊝Ꮉᵗ৻,๟ᦼ⊛ߥ᳓ᐔᄖജࠍฃߌࠆ᳓ߩ౉ߞߚ ౞╴࠲ࡦࠢߩ㕖✢ᒻᝄേᔕ╵_{,ᣣᧄᑪ▽ቇળ᭴ㅧ} ♽⺰ᢥ㓸_{,╙ 74 Ꮞ,╙ 642 ภ,pp.1461-1468 (2009).} 12) ⊝Ꮉᵗ৻,⥄↱⴫㕙ࠍᜬߟࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢ ᕈኈེߣߩᄢᄌᒻㅪᚑ໧㗴ߩ⋥ធ⸃_{,ᣣᧄᑪ▽ቇ} ળ਻Ꮊᡰㇱ_{49-1,pp.209-212 (2010).} 13) ⊝Ꮉᵗ৻,⥄↱⴫㕙ࠍᜬߟࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢ ᕈኈེߣߩᄢᄌᒻㅪᚑ໧㗴ߩ⋥ធ⸃_{,ᣣᧄᑪ▽ቇ} ળቇⴚ⻠Ṷ᪪᭎㓸_{B-1,251-252 (2010).} 14) ⊝Ꮉᵗ৻,⥄↱⴫㕙ࠍᜬߟࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢ ᕈኈེߣߩᄢᄌᒻㅪᚑ໧㗴ߩ⸃ᨆ㧘౞╴࠲ࡦࠢ ߩᄢᄌᒻേ⊛ㅪᚑ໧㗴ߩቯᑼൻ_{,ᣣᧄᑪ▽ቇળ਻} Ꮊᡰㇱ_{50-1,pp.345-348 (2011).} 15)ጊᧄᙗม,⊝Ꮉᵗ৻㪃ᶋደᩮࡐࡦ࠷࡯ࡦߩੑᰴࡕ࡯ ࠼౒ᝄߦࠃࠆᬦ౞ൻᄌᒻߦ㑐ߔࠆᬌ⸛_,ࠪࡦࠣ ࡞࠺࠶ࠠဳᶋደᩮࠍ᦭ߔࠆ౞╴ᶧ૕⾂ᮏߩ㕖✢ ᒻࠬࡠ࠶ࠪࡦࠣ⸃ᨆ_{,ᣣᧄᑪ▽ቇળ᭴ㅧ♽⺰ᢥ} 㓸_{,Vol.77,No.671,pp.35-44 (2012).} 16)⊝Ꮉᵗ৻,ੑᰴరߩ⥄↱⴫㕙ࠍᜬߟࡐ࠹ࡦࠪࡖ࡞ ᵹ૕ߣᒢᕈኈེߣߩㅪᚑ໧㗴ߩᢙ୯⸃ᨆ_,㣮ఽ ፉᄢቇᎿቇㇱ⎇ⓥႎ๔_{,54 ภ,pp.7-25 (2012).} 17)⊝Ꮉᵗ৻,ᐔ᧼ᦛߍ྾ⷺᒻⷐ⚛㧘߅ࠃ߮౞╴ࠪࠚ ࡞_{4 ▵ὐⷐ⚛ߩ㜞♖ᐲൻ,ᣣᧄᑪ▽ቇળ,਻Ꮊᡰ} ㇱ_{52㨯1,265-268 (2013).} 18)⊝Ꮉᵗ৻,⥄↱⴫㕙ࠍᜬߟࡐ࠹ࡦࠪࡖ࡞ᵹ૕ߣᒢ ᕈኈེߣߩᄢᄌᒻㅪᚑ໧㗴ߩ⸃ᨆ _{,౞╴࠲ࡦࠢ} ߩಽᢙ⺞ᵄᔕ╵ߩ⸃ᨆ_{,ᣣᧄᑪ▽ቇળ,਻Ꮊᡰㇱ} 52㨯1,269-272 (2013).

19)⊝Ꮉᵗ৻,DYNAMIC INTERACTIVE BEHAVIOR BETWEEN A POTENTIAL FLUID AND

ELASTIC CONTAINER IN LARGE

DEFORMATIONS,Part1,ᣣᧄᑪ▽ቇળ᭴ㅧ♽⺰ ᢥ㓸_{,╙ 78 Ꮞ,╙ 690 ภ,pp.1439-1448 (2013)} 20)Novozhilov,Foundation of theNonlinear Theory of

Elasticity,Graylock (1971)

21) C.A.ࡉ࡟ࡆࠕ,J.J.ࠦ࠽࡯,᦭㒢ⷐ⚛ᴺߩၮᧄߣᔕ↪㧘 ࡉ࡟ࠗࡦ࿑ᦠ,ᤘ๺ 55ᐕ(1980).

22) A.B.Sabir,Strain-Based Finite Elements for the Analysis of Cylinders with Holes and Normally Intersecting Cylinders,Nucleal Eng. And Design,vol.76,2,pp.111-120 (1983)

23) ᢙ୯ᵹ૕ജቇ✬㓸ᆔຬળ,⒖േႺ⇇ᵹࠇ⸃ᨆ㧘 ᧲੩ᄢቇ಴ ળ_(1995).