差し替え用PDFファイル

 604 ု⚝ߩᏅಽࡕ࠺࡞೙ᓮೣߦၮߠߊᵄേ೙ᓮ
Wave Control of Suspended Rope Based on Finite Difference Model
٤ ㈕ ዊ⯗㧔᧲ᵗᄢ㒮㧕 ᱜ ⷏ㇹ ቬ₵㧔᧲ᵗᄢ㧕
Xiao Lan ZHENG, Graduate School of Engineering, Toyo University, 2100 Kujirai, Kawagoe, Saitama
Muneharu SAIGO, Toyo University, 2100 Kujirai, Kawagoe, Saitama
This paper describes the wave control of suspended rope by using the control law based on a finite difference
(FD) model. The non-existing term in the boundary node equation due to the boundary condition is restored
by control, which is computed with the wave propagating solution of the interior node equation as if no
boundary existed. The wave propagating solution of the finite difference suspended rope system is newly
derived in this paper by introducing the new variable of the difference between adjacent displacements. A
similar process to the multiple pendulums system study (2) is applied to the finite difference rope system and
the displacement transfer function between adjacent node displacement in frequency domain and the
convolution integral kernel function in time domain have been obtained. The frequency analysis and
numerical time simulation have confirmed the wave control characteristics of the control law. The simulation
of the suspended rope system with load at the bottom has also been conducted and shown the controllability
superior to wave control of the non-homogeneous multiple simple pendulum system.
Key Words: ု⚝㧘ᵄേ೙ᓮ㧘ᝄࠇᱛ߼೙ᓮ㧘ࠢ࡟࡯ࡦࡠ࡯ࡊ 㧝㧚ߪߓ߼ߦ
ㄭᐕ㧘ࡠ࡯ࡊ߿ߪࠅߥߤߩ৻ᰴర᭴ㅧ૕ߩᵄേ೙ᓮ⎇ⓥ
߇⌕⋡ߐࠇߡ޿ࠆ㧚㐳ᄢ᭴ㅧ૕ߩᝄേࠛࡀ࡞ࠡ࡯ࠍ᭴ㅧ૕
ߩ๟ㄝߢലᨐ⊛ߦๆ෼ߔࠆߦߪ㧘ᝄേࠍቯ࿷ᵄߣߒߡᛒ߁
ߩߢߪߥߊᵄേવ᠞․ᕈࠍ೑↪ߒߡࠛࡀ࡞ࠡ࡯ๆ෼ߔࠆߎ
ߣ߇᦭ലߢ޽ࠆ㧚⷏ㇹࠄߪᵄേ೙ᓮߩታ↪ᯏེ߳ߩㆡ↪ࠍ
⋡ᮡߣߒߡ㧘 ᰴర᭴ㅧ૕ߩ೙ᝄߦ઒ᗐ♽ߩࠝࡦ࡜ࠗࡦࠪ
ࡒࡘ࡟࡯࡚ࠪࡦᛛᴺࠍ↪޿ࠆᵄേવ᠞․ᕈࠍ೑↪ߒߚ೙ᓮ
ᴺߩ⎇ⓥߦขࠅ⚵ࠎߢ߈ߚ㧚ߘߩㆡ↪଀ߦᄙ㊀නᝄሶ ု
♽ߩᡰᜬὐടㅦᐲ೙ᓮᴺ߇޽ࠆ 㧚ߐࠄߦ㧘ᄙ㊀නᝄሶ♽
ߢࠝࡦ࡜ࠗࡦࠪࡒࡘ࡟࡯࡚ࠪࡦࠍᔅⷐߣߒߥ޿ᵄേવ᠞⸃
ࠍ⠨᩺ߒߡ೙ᝄᕈࠍะ਄ߐߖ㧘ᄙ㊀නᝄሶ♽ࠍ੺ߒߚࠢ࡟
࡯ࡦࡠ࡯ࡊ♽ߩᣂߒ޿೙ᝄ᭴ㅧߩឭ᩺ࠍⴕߞߡ޿ࠆ 㧚ߒ
߆ߒߥ߇ࠄ㧘ᄙ㊀නᝄሶࠍ੺ߒߚษ⩄ࡠ࡯ࡊ ု♽ߪ㕖ဋ
⾰♽ߢ޽ࠆߚ߼ࡠ࡯ࡊ㐳߇㐳ߊߥࠆߣ೙ᝄߦᤨ㑆߇߆߆ࠆ
ߎߣ߽᣿ࠄ߆ߦߒߡ޿ࠆ㧚
ߘߎߢ㧘ᧄ⎇ⓥߪ㧘ᄙ㊀නᝄሶ♽ߩᵄേવ᠞ߦၮ␆ࠍ⟎
ߊߩߢߪߥߊ㧘 ုࡠ࡯ࡊࠍ⋥ߦᵄേๆ෼೙ᓮࠍⴕ߁႐ว
ߩ ုࡠ࡯ࡊ♽ߩ೙ᓮലᨐࠍ㧘නᝄሶ♽ࠍ੺ߒߚࡠ࡯ࡊ♽
ߩ೙ᝄലᨐߣᲧセߔࠆߎߣߦࠃߞߡ᣿ࠄ߆ߦߔࠆߎߣࠍ⋡
⊛ߣߒߡ޿ࠆ㧚 ုࡠ࡯ࡊߩ࿕ቯ਄┵ㄭறߢᵄേ೙ᓮࠍⴕ
߁ߚ߼㧘 ုࡠ࡯ࡊߩᏅಽㄭૃㆇേᣇ⒟ᑼࠍዉ߈㧘Ⴚ⇇ㄭ
ற㧔Ⴚ⇇▵ὐ㧕ߢࠗࡦࡇ࡯࠳ࡦࠬᢛว೙ᓮߦࠃࠆᵄേๆ෼
೙ᓮࠍⴕߞߡ޿ࠆ㧚
㧞㧚ㆇേᣇ⒟ᑼߣ೙ᓮೣ
࿑ 1(a)ߦ␜ߔ ု⚝ߩㆇേᣇ⒟ᑼߪᰴᑼߢਈ߃ࠄࠇࠆ
ߔߥࠊߜ㧘ᑼ(4)ฝㄝ߇೙ᓮ㗄ߢ޽ࠆ㧚Ꮕಽࡕ࠺࡞⸘▚ߢߪ
ᄌ૏ࠍ೙ᓮ㊂ߣߒߡᛒ߃߫ࠃ޿㧚ಽᏓਸ਼ᢙ♽⸘▚ߢߪᑼ(4)
ߩᏀㄝࠍ㧘න૏㐳ߐ޽ߚࠅ⾰㊂ࠍ P ߣߒߡ㧘 ( P'z ) yn ߣߥ
ࠆࠃ߁ߦᢛℂߒߚߣ߈ߩ 'z ඙㑆ߦ૞↪ߔࠆ೙ᓮജߣߒߡ
ᛒ߃߫ࠃ޿㧚
ᰴߦ઒ᗐ⊛ᄌ૏ yn 1 ߩ⸘▚ᣇᴺࠍ␜ߔ㧚ᑼ(4)ߪౝㇱ▵ὐ
ᣇ⒟ᑼߣหߓߢ޽ࠆߩߢౝㇱ▵ὐᣇ⒟ᑼߩᵄേવ᠞⸃ࠍ↪
޿ࠆ㧚ᵄേવ᠞⸃ߪએਅߩࠃ߁ߦ᳞߼ࠄࠇࠆ㧚
╙ m ▵ὐㆇേᣇ⒟ᑼߣ╙ m 1 ▵ὐㆇേᣇ⒟ᑼߩᏅࠍߣ
ࠅ ym ym 1 bm ߣ⟎޿ߡ࡜ࡊ࡜ࠬᄌ឵ L >bm @ Bm ߔࠆߣ㧘
2
mBm 1 ª 2 s Z0 º m 1 Bm m 2 Bm 1
¬
¼
Z02 g 2m 1 2'z ᑼ(5)ߩ৻⥸⸃ࠍ
Bm s Z0 1
m
2m 1 J s Z0 J s Z0 ª s 2 4Z02 s º
L1 ¬ªJ s Z0 ¼º
2 J 2 2Z0t t " (8)
ߎߎߢ J 2 ߪ૏ᢙ 2 ߩ╙৻⒳ࡌ࠶࠮࡞㑐ᢙ㧚
O
0 ࠃࠅ㧘Ⴚ⇇▵ὐᣇ⒟ᑼߪ,
0 " (3) g
y
ym1
ym
ym1
'z
Gravity
y
wy wx
Tension
y2
y1
P g l x dx x
(a)
(b)
Fig. 1 Suspended rope system
(a): Distributed system (b): Finite difference system
茨城講演会講演論文集 （共催　日本機械学会関東支部 ・ 精密工学会 ・ 茨城大学， 2013-9-6， 日立）
－ 41 －
0
yn
yn1
P g l x dx
z
yn1
y
x
l
ᑼ(3)ߢႺ⇇᧦ઙࠍ⠨ᘦߒߥ޿ᑼ(4)߇ታ⃻ߔࠇ߫Ⴚ⇇߇㒰
෰ߐࠇή㒢᭴ㅧ߇ታ⃻ߔࠆ㧚
2'z g yn [2 2n yn 2n 1 yn 1 ] 2n 1 yn 1 " (4) 2
4 Z02 " (7)
¬
¼
ࠍᓧࠆ㧚ᑼ(7)ࠍㅒ࡜ࡊ࡜ࠬᄌ឵ߔࠆߣ㧘
g ª¬ z w 2 y wz 2 wy wz º¼ 0 "" (1)
ᑼ(1)ࠍ࿑ 1(b)ߦ␜ߔᏅಽߦࠃߞߡᏅಽㄭૃᑼߢ⴫ߔߣ,
2'z g ym [ 2m 1 ym 1
2 2m ym 2m 1 ym 1 ] 0 m z n " (2)
2'z g yn [2 2n yn 2n 1 yn 1 ]
" (6)
ߣ઒ቯߔࠆߣ㧘․ᕈᩮ
w 2 y wt 2
Ⴚ⇇᧦ઙ yn 1
0 䇭" (5)
g