Development of Thin-Type Four-Axis Force/Moment Sensor for Robot Hand

(1)

Development of Thin-Type Four-Axis Force/Moment Sensor for Robot Hand

Yuichiro HAYASHI*, Nobutaka TSUJIUCHI**, Takayuki KOIZUMI*** and Hiroko OSHIMA****

(Received April 17, 2009)

Multi-axis force/moment sensor, which can measure force and moment, is useful for robotic force control. However, the smallest multi-axis force/moment sensor has excess height, which in effect worsens the stability of tasks. With the reason, it is expected to develop a thin type multi-axis force/moment sensor, which enables robot hand to grasp object, as if it were human. In the present study, we aim to develop a thin type four-axis force/moment sensor, which can be mounted to finger cushion direction. Firstly, we designed four-axis force/moment sensor and optimized positions, placing strain gauges from finite element analysis. Secondly, we applied these determined positions from simulation model and obtained sufficient sensitivities in each axial force and twisting moment around z-axis, as a result of sensitivity evaluation. Finally, we made a production model of four-axis force/moment sensor and performed characteristics test by applying determined positions. As results in the present study, we obtained sufficient sensitivities in each axial force and twisting moment around z-axis by sensitivity evaluation of experiments, and showed the effectiveness of the developed sensor.

-G[YQTFU robot hand, four-axis force/moment sensor, strain gauge, optimization, sensitivity

ࠠ࡯ࡢ࡯࠼ࡠࡏ࠶࠻ࡂࡦ࠼㧘㧠ゲജⷡ࠮ࡦࠨ㧘߭ߕߺࠥ࡯ࠫ㧘ᦨㆡൻ㧘ᗵᐲ

ࡠࡏ࠶࠻ࡂࡦ࠼↪⭯ဳ4ゲജⷡ࠮ࡦࠨߩ㐿⊒

ᨋ␭৻㇢㧘ㄞౝિᅢ㧘ዊᴰቁਯ㧘ᄢፉ⵨ሶ

ߪߓ߼ߦ

⃻࿷㧘ࡠࡏ࠶࠻ߦ㑐ߔࠆ᭽ޘߥ⎇ⓥ߇ㅴࠎߢ޿ࠆ㧚

↥ᬺ↪ࡠࡏ࠶࠻¹⁾߇ਥߢ޽ࠆ߇㧘ੱဳߩੑ⿷ᱠⴕࡠ ࡏ࠶࠻²⁾ߥߤ㔛ⷐߦᔕߓߚࡠࡏ࠶࠻ߩ⎇ⓥ߅ࠃ߮㐿

⊒߽ㅴ߼ࠄࠇߡ޿ࠆ㧚ਛߢ߽ੱ㑆ߩᚻࠍᮨᡆߒߚേ

૞ࠍน⢻ߣߔࠆᄙᜰࡠࡏ࠶࠻ࡂࡦ࠼ࠍᜬߟࡠࡏ࠶࠻

ߪᄙ޿㧚ߘࠇࠄࠍ↪޿ߡ㧘ኻ⽎‛ࠍ⪭ߣߐߕߦߟ߆

ߺ⛯ߌࠆᛠᜬേ૞߿㧘‛૕ߩ૏⟎߹ߚߪᆫ൓ࠍᄌൻ ߐߖࠆᠲࠅേ૞ߥߤࠍታ⃻ߔࠆߚ߼ߩ⎇ⓥ߇ᄙߊ߅ ߎߥࠊࠇߡ޿ࠆ^{3, 4)}㧚ᛠᜬ߿ᠲࠅߥߤߩേ૞ࠍߔࠆ㓙 ߦߪࡠࡏ࠶࠻ߩജ೙ᓮࠍ߅ߎߥ߁ߚ߼ߦ㧘ᜰవ߇ኻ

⽎‛ߣߩធ⸅㕙߆ࠄฃߌࠆജ߿ࡕ࡯ࡔࡦ࠻ߩ⸘᷹߇ ᔅⷐߣߥࠆ㧚

ࡠࡏ࠶࠻ࡂࡦ࠼ߩᜰవ߇ฃߌࠆᄖജࠍ⸘᷹ߔࠆⵝ

* Department of Mechanical Engineering, Doshisha University, Kyoto

Telephone: +81-774-65-6488, Fax: +81-774-65-6488, E-mail: etj1302@mail4.doshisha.ac.jp

** Department of Mechanical Engineering, Doshisha University, Kyoto Telephone/Fax: +81-774-65-6493, E-mail: ntsujiuc@mail.doshisha.ac.jp

*** Department of Mechanical Engineering, Doshisha University, Kyoto Telephone/Fax: +81-774-65-6492, E-mail: tkoizumi@mail.doshisha.ac.jp

**** Department of Mechanical Engineering, Doshisha University, Kyoto Telephone/Fax: +81-774-65-6488, E-mail: sk107645@mail.doshisha.ac.jp

(2)

⟎ߣߒߡઍ⴫⊛ߥ߽ߩߪᄙゲജⷡ࠮ࡦࠨߢ޽ࠆ㧚ⶄ ᢙߩゲᣇะߦหᤨߦ૞↪ߔࠆᄖജࠍ⸘᷹ߒ㧘ࡠࡏ࠶

࠻ࡂࡦ࠼ߦⵝ⌕น⢻ߥ᭽ޘߥᒻ⁁ߩᄙゲജⷡ࠮ࡦࠨ ߇㐿⊒ߐࠇߡ޿ࠆ㧚ߘߩᄙߊߪ⿠ᱡㇱߩ⴫㕙ߦ⾍ߞ ߚ߭ߕߺࠥ࡯ࠫߩ㔚᳇ᛶ᛫ᄌൻ߆ࠄᄖജࠍᬌ಴ߔࠆ

߽ߩߢ޽ࠅ㧘ࡠࡏ࠶࠻ߦജᗵⷡࠍઃਈߔࠆ㊀ⷐߥ⸘

᷹ⵝ⟎ߣ޿߃ࠆ^{5, 6)}㧚

ߒ߆ߒ,⃻࿷Ꮢ⽼ߐࠇߡ޿ࠆࡠࡏ࠶࠻ࡂࡦ࠼ߩᜰ వ↪6ゲജⷡ࠮ࡦࠨߪ㧘ᦨዊߩ߽ߩߢ⋥ᓘ12[mm]㧘㜞ߐ15[mm]ߩ౞ᩇဳߢ޽ࠅ㧘ᨵエⵍⷒ᧚ࠍ࠮ࡦࠨߩ వ┵ㇱߦ߆߱ߖߡ૶↪ߔࠆߚ߼㧘ᜰవ߇ඨ⃿ᒻ⁁ߣ ߥࠆ⁷⁾㧚ߎߩ࠮ࡦࠨߪ᭴ㅧ਄㧘ඨ⃿ᒻߩవ┵ㇱߦฃ ߌߚᄖജએᄖߪ⸘᷹ߢ߈ߕ㧘⸳⟎ᣇะ߽છᗧߦ᳿ቯ ߢ߈ߥ޿㧚ߒߚ߇ߞߡ㧘࠮ࡦࠨߩవ┵ㇱ߇ኻ⽎‛ߩ 㕙ߦኻߒߡု⋥ߦㄭ޿ⷺᐲߢធ⸅ߔࠆᔅⷐ߇޽ࠅ㧘 ᛠᜬേ૞߿ᠲࠅേ૞ߥߤߩ቟ቯᕈ߇ૐਅߔࠆ㧚ࠃߞ ߡ㧘ੱ㑆߇ኻ⽎‛ࠍߟ߆߻ߥߤᜰߩ⣻ࠍ↪޿ߚᛠᜬ

േ૞ࠍน⢻ߣߔࠆߚ߼⁸⁾㧘ᜰߩ⣻ᣇะߦⵝ⌕ߒߡኻ

⽎‛߆ࠄฃߌࠆᄖജࠍ⸘᷹น⢻ߥ⭯ဳߩᄙゲജⷡ࠮

ࡦࠨߩ㐿⊒߇ᔅⷐߢ޽ࠆ㧚

ߘߎߢ㧘ᧄ⎇ⓥߢߪࡠࡏ࠶࠻ࡂࡦ࠼ߩᜰߩ⣻ᣇะ ߦⵝ⌕ߒߡ㧘ኻ⽎‛߆ࠄฃߌࠆᄖജࠍᬌ಴น⢻ߥ⭯

ဳߩ4ゲജⷡ࠮ࡦࠨߩ㐿⊒ࠍ⋡⊛ߣߔࠆ㧚㐿⊒ߔࠆ࠮

ࡦࠨࠍx, y, zᐳᮡⓨ㑆ߦ߅߈㧘਄㕙ߦኻߒߡߖࠎᢿ ᣇะࠍx, y^{ゲᣇะ㧘᛼࿶ᣇะࠍ}z^{ゲᣇะߣߒߚᤨ㧘} ᧄ࠮ࡦࠨߪx, y, z^{ゲᣇะߩ3ಽജ}F_x, F_y, F_z^ߣz^ゲ

࿁ࠅߦ૞↪ߔࠆࡕ࡯ࡔࡦ࠻M_z^ߩ⸘4ಽജࠍ⸘᷹น

⢻ߢ޽ࠆ㧚߹ߚ㧘ᣢሽߩ࠮ࡦࠨߣᲧセߒߡ⭯ဳߩᒻ

⁁ߣߒ㧘⸘᷹ේℂߣߒߡߪ⿠ᱡㇱߩ⴫㕙ߦ⾍ߞߚ߭

ߕߺࠥ࡯ࠫߩ㔚᳇ᛶ᛫ᄌൻ߆ࠄജߣࡕ࡯ࡔࡦ࠻ࠍᬌ

಴ߔࠆᣇᑼࠍណ↪ߒߡ޿ࠆ㧚ᧄ⎇ⓥߢߪ࠮ࡦࠨߩᒻ

⁁ࠍ⸳⸘ߒ᦭㒢ⷐ⚛⸃ᨆࠍ߅ߎߥ޿㧘߭ߕߺࠥ࡯ࠫ

ߩ⾍ࠅઃߌ૏⟎ࠍ᳿ቯߔࠆ㧚ߐࠄߦ㧘ታᯏࡕ࠺࡞ࠍ

⹜૞ߒ㧘ࠪࡒࡘ࡟࡯࡚ࠪࡦߦࠃߞߡ᳿ቯߒߚ⾍ࠅઃ

ߌ૏⟎ࠍㆡ↪ߒߡ․ᕈ⹜㛎ࠍ߅ߎߥߞߡ㧘㐿⊒ߒߚ

࠮ࡦࠨߩ᦭ലᕈࠍᬌ⸽ߔࠆ㧚

⭯ဳゲജⷡ࠮ࡦࠨ

࠮ࡦࠨߩ⸳⸘

㐿⊒ߔࠆ4ゲജⷡ࠮ࡦࠨߩ઀᭽⋡ᮡ୯ࠍ⴫1ߦ␜ߔ㧚

ᧄ᧪㧘ᄙゲജⷡ࠮ࡦࠨߣߒߡ㐿⊒ߐࠇߚ߽ߩߪ6ゲജ

ⷡ࠮ࡦࠨ߇৻⥸⊛ߢ޽ࠆ߇㧘ߎߎߢߪ⥄↱ⓨ㑆ߦ߅ ߌࠆኻ⽎‛ߩ㊀ᔃᐳᮡὐࠍ⸘᷹ߔࠆߎߣߪ⋡⊛ߣߒ ߡ޿ߥ޿㧚ߘߩߚ߼㧘᛼࿶ᣇะߩജ߅ࠃ߮Ṗࠅߦࠃ ࠆ៺ᡂജએᄖߦኻ⽎‛߇ߕࠇ⪭ߜࠆ㓙ߩᝳࠅࡕ࡯ࡔ ࡦ࠻ࠍ⸘᷹ኻ⽎ߣߒߚ4ゲജⷡ࠮ࡦࠨࠍ㐿⊒ߔࠆ㧚

߹ߚ, ⴫1ߦ␜ߔࠃ߁ߦ࠮ࡦࠨߩ㜞ߐߪ5[mm]ߢ޽

ࠅ㧘Ꮢ⽼ߐࠇߡ޿ࠆᦨዊߩᄙゲജⷡ࠮ࡦࠨߣᲧセߒ ߡ⭯ဳߩᒻ⁁ߣ޿߃ࠆ㧚ߒ߆ߒ, ߎߩ㜞ߐߢߪ࠮ࡦ ࠨߩ⿠ᱡㇱ߇ዊဳൻߒߡߒ߹߁ߚ߼㧘ജⷡ࠮ࡦࠨߦ ߅ߌࠆઍ⴫⊛ߥ⿠ᱡㇱߩᒻ⁁ߢ޽ࠆ౞ᩇဳ⁹⁾߽ߒߊ ߪචሼࡆ࡯ࡓဳ^{10, 11)}ߩ߁ߜ, ࿑1ߦ␜ߔචሼࡆ࡯ࡓ

ဳߩ᭴ㅧࠍㆬᛯߔࠆ㧚ᧄ࠮ࡦࠨߪ࿑1ߦ␜ߔ4ᧄߩ᪞

ߩ஥㕙ߦ⾍ߞߚ߭ߕߺࠥ࡯ࠫߩ㔚᳇ᛶ᛫ᄌൻ߆ࠄ࠮

ࡦࠨਛᔃߦ૞↪ߒߚᄖജࠍᬌ಴ߔࠆ㧚߭ߕߺࠥ࡯ࠫ

ߪᛶ᛫୯120[ǡ]㧘ࠥ࡯ࠫ₸1.9, ⥄Ꮖ᷷ᐲ⵬ఘ▸࿐0 㨪150[͠]㧘㐳 ߐ1.6[mm], ᏷1.2[mm], ࠥ ࡯ ࠫ 㐳 0.2[mm]ߢ޽ࠆනゲᑼߩ౒๺㔚ᬺ⵾▗߭ߕߺࠥ࡯ࠫ

KFR-02N-120-C1-11-N10C2ࠍ↪޿ࠆ㧚ᧄ࠮ࡦࠨߦ૶

↪ߔࠆ߭ߕߺࠥ࡯ࠫߪ⸘16ᨎߢ޽ࠆ㧚

ታ㓙ߦ⵾૞ߔࠆ࠮ࡦࠨߩᒻ⁁ࠍ࿑2ߦ␜ߔ㧚࿑2ࠃ ࠅ,⩄㊀ࠍ࠮ࡦࠨߩਛᔃㇱߦኻߒߡဋ╬ߦ⽶⩄ߔࠆ ߚ߼ࡇࡦࠍਛᔃㇱߦขࠅઃߌߚ᭴ㅧߣߥߞߡ޿ࠆ㧚

࠮ࡦࠨᧄ૕㧘ࡇࡦߩ᧚ᢱߪߘࠇߙࠇࠕ࡞ࡒ࠾࠙ࡓว

㊄A2024㧘ࠬ࠹ࡦ࡟ࠬSUS304ߢ޽ࠆ㧚

⸘᷹ේℂ

ߎߩ࠮ࡦࠨߩ․ᓽߪ㧘ว⸘8ᨎߩ߭ߕߺࠥ࡯ࠫߦࠃ ߞߡ᭴ᚑߐࠇࠆ4⒳㘃ߩࡂ࡯ࡈࡉ࡝࠶ࠫ࿁〝ߩ಴ജ ࠍ⚵ߺวࠊߖߡF_x, F_y,M_zߦኻᔕߔࠆ߭ߕߺࠍᬌ಴

น⢻ߥߎߣߢ޽ࠆ㧚߭ߕߺࠥ࡯ࠫࠍ⾍ࠅઃߌࠆ㕙߅ ࠃ߮ࡉ࡝࠶ࠫ࿁〝ߩ᭴ᚑ࿑ࠍ࿑3ߦ␜ߔ㧚ߎߎߢ㧘࿑

ਛߩ߭ߕߺࠥ࡯ࠫA1, A2, B1, B2, C1, C2, D1, D2ߪߖ ࠎᢿ߭ߕߺࠍ⸘᷹ߔࠆߚ߼᪞ߩਛ┙ゲߦኻߒߡᢳ߼

45[deg]ᣇะߦ⾍ߞߚ߭ߕߺࠥ࡯ࠫߢ޽ࠅ㧘A3, A4,

B3, B4, C3, C4, D3, D4ߪゲᣇะജࠍ⸘᷹ߔࠆߚ߼ゲ ᣇะߦኻߒߡု⋥ߦ⾍ߞߚ߽ߩߢ޽ࠆ㧚߭ߕߺߩṶ

▚ᑼߪ㧘࿑3ߦ␜ߔࡉ࡝࠶ࠫ࿁〝⇟ภ1㨪5ߦኻᔕߔࠆ

߭ߕߺߩ಴ജࠍH₁,H₂,H₃,H₄,H₅, ^{ฦ⩄㊀ᚑಽߦኻᔕ} ߔࠆ߭ߕߺߩ಴ജࠍ

z z y

x F F M

F H H H

H , , , ^{ߣߔࠆߣ㧘ᰴᑼ}

(3)

ߢ⴫ߐࠇࠆ㧚

2

1 H

H

Fx (2-1)

4

3 H

H

Fy (2-2) H5

HFz (2-3) 0

. 2 / ) (H₁ H₂ H₃ H₄

H

Mz (2-4)

ߎߎߢ㧘ฦ߭ߕߺࠥ࡯ࠫߦ↢ߓࠆ߭ߕߺࠍH_A1,...,

HD4^ߣߔࠆߣ㧘 B3 D4

1 H H

H (2-5)

B4 D3

2 H H

H (2-6)

A3 C4

3 H H

H (2-7)

A4 C3

4 H H

H (2-8)

D2 D1 C2 C1 B2 B1 A2 A1

5 H H H H H H H H

H (2-9)

ߩࠃ߁ߦߥࠆ㧚

ᧄ࠮ࡦࠨߪ4ᧄߩ᪞ߦࠃࠆචሼࡆ࡯ࡓᣇᑼࠍណ↪

ߒߡ޿ࠆߚ߼㧘ฦゲᣇะ⩄㊀߇⽶⩄ߐࠇߚᤨߦ⸘᷹

ߔࠆኻ⽎એᄖߩ⩄㊀ᚑಽߦኻߒߡ಴ജ߇⊒↢ߔࠆ⋧

੕ᐓᷤ߇⿠ߎࠆ㧚ᑼ(2-1)́(2-9)ߦ␜ߒߚฦ߭ߕߺߩ Ṷ▚ᑼߪ㧘ߎࠇߦࠃߞߡ↢ߓࠆᐓᷤᚑಽࠍṶ▚ߦࠃ ࠅᛂߜᶖߔߎߣ߇ߢ߈ࠆࠃ߁ߦ⸳ቯߒߡ޿ࠆ㧚

Table 1. Technology Target Value.

Fig.1. Overhead View of Four-Axis Force/Moment Sensor.

Fig.2. Structure of the Production Model.

Fig.3. Position to Paste Strain Gauges and Connecting Diagram on the Four-Axis Force/Moment Sensor.

ജⷡ࠮ࡦࠨߩ⹏ଔᣇᴺ

ᧄ⺰ߢߪജⷡ࠮ࡦࠨߩᕈ⢻ࠍቯ㊂⊛ߦ⹏ଔߔࠆߚ

߼㧘ౝጊ¹²⁾߅ࠃ߮ਛ᧛¹³⁾ߦࠃߞߡឭ᩺ߐࠇߚએਅߦ

␜ߔ⹏ଔᣇᴺࠍ↪޿ࠆ㧚࠮ࡦࠨߦ૞↪ߒߚᄖജ f ^ߣ㧘

૞↪ߒߚᄖജߦኻᔕߒߡ಴ജߔࠆ߭ߕߺİ^{ߩ㑐ଥߪ㧘} ᐓᷤᚑಽ߇✢ᒻߢ޽ࠆߣ઒ቯߔࠆߣ㧘߭ߕߺࠦࡦࡊ Geometric

Dimension

Hysteresis Material Nonlinearity

Rating Capacity

100 [N]

0.1 [N䍃m]

Within 2 [%RO]

A2024 Within 2 [%RO]

z y

x F F

F, ,

20 [mm](W)㬍20 [mm](D)㬍5 [mm](H)

Mz

Geometric Dimension

Hysteresis Material Nonlinearity

Rating Capacity

100 [N]

0.1 [N䍃m]

Within 2 [%RO]

A2024 Within 2 [%RO]

z y

x F F

F, ,

20 [mm](W)㬍20 [mm](D)㬍5 [mm](H)

Mz

Bridged Circuit 1

Perpendicular Direction䋺F_z

Bridged Circuit 4 Bridged Circuit 2

C4 A3 R R

C3 A4 R R

B4 D3

R R

D4 B3

R R

A1

D2 C1 A2

D1

B2

C2 B1

ei

ei e_i ei

E

E E

E

E Fx

A4

D2 A3 A1

D1 D4 D3

A2

C2 C1 C4 C3

B3 B4

B1 B2 Mz

Fy

Fx

Bridged Circuit 1

Perpendicular Direction䋺F_z

C4 A3 R R C4 A3

R R

C3 A4 R R C3 A4

R R B4 D3

R R

D4 B3

R R

D4 B3

R R

A1

D2 C1 A2

D1

B2

C2 B1 A1

D2 C1 A2

D1

B2

C2 B1

ei

ei e_i ei

E

E E

E

E Fx

A4

D2 A3 A1

D1 D4 D3

A2

C2 C1 C4 C3

B3 B4

B1 B2 Mz

Fy

Fx

Pin

(4)

࡜ࠗࠕࡦࠬⴕ೉Cࠍ↪޿ߡᰴᑼߢ⴫ߐࠇࠆ㧚

İ Cf (3-1)

ߚߛߒ㧘

>

xj yj zj zj

@

^T

j F F F M

f (3-2)

>

Fj Fj Fj M j

@

^T

j H _x H _y H _z H _z

İ (3-3)

ߣߔࠆ㧚ߎߎߢ, İ

>

İ₁ İ_n

@

^,f

>

f₁ f_n

@

^,

, ..., ,

1 n

j nߪ࠺࡯࠲ᢙߣߔࠆ㧚߹ߚ㧘ᚑಽߏߣߩ න૏ߩขࠅᣇ߿ᰴరߩ㆑޿ߦࠃࠆᓇ㗀ࠍឃ㒰ߔࠆߚ

߼ߦ㧘İߪ⸘᷹ߒߚ߭ߕߺߩᦨᄢ୯㧘f ^ߪജ߅ࠃ߮

ࡕ࡯ࡔࡦ࠻ߩᦨᄢ୯ߦࠃߞߡฦᚑಽߏߣߦᱜⷙൻߒ ߡİ, f ^ߣߔࠆߣᑼ(3-1)ߪᰴᑼߢ⴫ߖࠆ㧚

İ Cf (3-4) , f

İ ࠍ↪޿ߡዉ಴ߐࠇࠆ߭ߕߺࠦࡦࡊ࡜ࠗࠕࡦࠬ

ⴕ೉C^{ࠍ⹏ଔߦ↪޿ࠆ㧚}Cࠍ᳞߼ࠆ㓙ߪ㧘૞↪ߒߚ ᄖജ߅ࠃ߮߭ߕߺߩ࠺࡯࠲ࠍߘࠇߙࠇ4⒳㘃એ਄ਗ ߴߚⴕ೉ࠍᱜⷙൻߒߚ f ^ߣİ^{ࠍ↪޿ࠆ㧚}İ^ߦ f ^ߩ৻

⥸ൻㅒⴕ೉ fࠍฝ߆ࠄដߌࠆߎߣߦࠃࠅ㧘C^ࠍ᳞

߼ࠆߎߣ߇ߢ߈ࠆ㧚Cߦ߅ߌࠆฦⴕࡌࠢ࠻࡞ߩࡁ࡞

ࡓ C_i ^{ࠍ߭ߕߺࠥ࡯ࠫᗵᐲ㧘}C^{ߩᦨዊ․⇣୯}V_min^ࠍ ജ ⷡ ᗵᐲ 㧘ᦨ ᄢ ․⇣ ୯ࠍ ᦨ ዊ․ ⇣୯ ߢ ഀߞ ߚ୯

min max/V

V ^ࠍ᧦ઙᢙcond.Cߣቯ⟵ߒ㧘ߘࠇߙࠇ⹏ଔ ߦ↪޿ࠆ㧚߭ߕߺࠥ࡯ࠫᗵᐲߣߪ㧘ജ߿ࡕ࡯ࡔࡦ࠻

ߦኻߔࠆ୘ޘߩ߭ߕߺࠥ࡯ࠫߩᗵᐲࠍ⴫ߒ㧘ജⷡᗵ ᐲߣߪ߭ߕߺࠥ࡯ࠫߩ಴ജ߇ో૕ߣߒߡ3ゲᣇะߩ ജ߿ゲ࿁ࠅߩࡕ࡯ࡔࡦ࠻ࠍචಽߦ฽ࠎߢ޿ࠆ߆ุ߆ ࠍ⹏ଔߔࠆ߽ߩߢ޽ࠆ¹³⁾㧚᧦ઙᢙߦߟ޿ߡߪ㧘

min max 0

0 0

V V H

H ^d

' d'F F

(3-5)

ߩ㑐ଥ߇ᚑ┙ߔࠆߎߣߦࠃࠅ㧘⸘᷹ߒߚ߭ߕߺ H0^߆

ࠄ૞↪ߒߚᄖജF₀ࠍ▚಴ߒߚ႐ว㧘ᄖജߦ߅ߌࠆ⺋

Ꮕ'Fߩഀว߇㧘⸘᷹ߒߚ߭ߕߺߦ߅ߌࠆ⺋Ꮕ'H^ߩ

ഀวߦኻߒߡ㧘ᦨᄢcond.C୚߹ߢ᜛ᄢߐࠇࠆߎߣࠍ

⴫ߔ㧚ജⷡ࠮ࡦࠨ߇ℂᗐ⊛ߥ᭴ㅧࠍ᦭ߔࠆ႐ว㧘ജ ߩฦᚑಽࠍ⁛┙ߒߡ⸘᷹ߢ߈㧘ᗵᐲ߽╬ߒߊߥࠆ㧚 ߒߚ߇ߞߡ㧘Cߪන૏ⴕ೉ߣߥࠅ㧘߭ߕߺࠥ࡯ࠫᗵ ᐲ㧘ജⷡᗵᐲ߅ࠃ߮᧦ઙᢙߪ1ߣߥࠆ¹⁴⁾㧚

߭ߕߺࠥ࡯ࠫ⾍ࠅઃߌ૏⟎ߩᦨㆡൻ

߭ߕߺࠥ࡯ࠫᑼജⷡ࠮ࡦࠨߦ߅޿ߡ㧘ฦゲᣇะߩ

ᗵᐲ߇੕޿ߦ╬ߒߊ㧘߆ߟ㜞ߊߥࠆࠃ߁ߦ߭ߕߺࠥ

࡯ࠫߩ⾍ࠅઃߌ૏⟎ࠍ᳿ቯߔࠆ૞ᬺߪਇนᰳߢ޽ࠆ㧚 ᧄ┨ߢߪ᦭㒢ⷐ⚛⸃ᨆߦࠃࠆ⸃ᨆ⚿ᨐࠍ↪޿ߡ㧘߭

ߕߺࠥ࡯ࠫߩ⾍ࠅઃߌ૏⟎ᐳᮡࠍᦨㆡൻߔࠆ㧚ߥ߅㧘 ᓟߦ╙ 5.6 ▵ߢㅀߴࠆࠃ߁ߦ೰ᕈߪ㒢⇇୯ߦㄭ޿ߚ

߼㧘ᧄ⎇ⓥߢߪᒻ⁁ᦨㆡൻߦߟ޿ߡ⠨ᘦߒߥ޿߽ߩ ߣߔࠆ㧚᦭㒢ⷐ⚛⸃ᨆߦߪ㧘᭴ㅧ⸃ᨆ࠰ࡈ࠻࠙ࠚࠕ 㧔UGS PLM ࠰ ࡝ ࡘ ࡯ ࠪ ࡚ ࡦ ␠ ⵾ NX Nastran,

Ver.4.1㧕ࠍ↪޿ࠆ㧚ജⷡ࠮ࡦࠨߩ᦭㒢ⷐ⚛ࡕ࠺࡞ߪ

࿑2ߦ␜ߔ߽ߩߣߒ, ⩄㊀᧦ઙߣߒߡߪ⴫1ߦ␜ߔ ቯᩰ⩄㊀ࠍ↪޿ࠆ㧚᦭㒢ⷐ⚛ࡕ࠺࡞ߪ▵ὐᢙ 2488, ⷐ⚛ᢙ8167ߢ޽ࠅ㧘⸃ᨆ᧦ઙߣߒߡࠕ࡞ࡒ࠾࠙ࡓߩ

❑ ᒢ ᕈ ଥ ᢙ72.4 [GPa]㧘 ࡐ ࠕ ࠰ ࡦ Ყ0.33^{㧘 ኒ ᐲ} 103

77 .

2 u [kg/m³]㧘ࠬ࠹ࡦ࡟ࠬߩ❑ᒢᕈଥᢙ193

[GPa]^{㧘ࡐࠕ࠰ࡦᲧ}0.3^㧘ኒᐲ7.93u10³[kg/m³]^ߣߒ㧘 4 㓈ߩࡏ࡞࠻ⓣࠍቢో᜔᧤ߒߚ႐วߦ߅޿ߡਛᄩㇱ ߦฦゲᣇะߩቯᩰ⩄㊀ࠍන⁛ߢ⽶⩄ߔࠆ߽ߩߣߔࠆ㧚

࿑3ߢ␜ߒߚA1, A2㧘B1, B2㧘C1, C2㧘D1, D2ࠍ

✚⒓ߒߡࠥ࡯ࠫ1㧘A3, A4㧘B3, B4㧘C3, C4㧘D3, D4 ࠍ✚⒓ߒߡࠥ࡯ࠫ2ߣߒ㧘⾍ࠅઃߌ૏⟎ߩᐳᮡࠍ᳿

ቯߔࠆ㧚࿑3ߦ߅ߌࠆ߭ߕߺࠥ࡯ࠫD1, D3, B4, B2 ࠍ⾍ࠆ㕙ࠍᱜ㕙߆ࠄ⷗ߚ࿑ࠍ࿑4ߦ␜ߔ㧚ߖࠎᢿ߭

ߕߺࠍ⸘᷹ߔࠆࠥ࡯ࠫ1ߪᨒߦㄭ޿ᄖ஥ߦ㧘ߘߩઁ

ߪਛᔃߦㄭ޿ౝ஥ߦ⾍ࠅઃߌࠆ߽ߩߣߔࠆ㧚߹ߚ㧘

߭ߕߺࠥ࡯ࠫߩਛὐߪ᪞ߩਛ┙ゲ਄ߦวࠊߖࠆ㧚⾍

ࠅઃߌน⢻ߥ▸࿐ౝߢ㧘᦭㒢ⷐ⚛⸃ᨆߦࠃߞߡ⩄㊀

⽶⩄ᤨߩㇱಽ߭ߕߺࠍ᳞߼ࠆ㧚⩄㊀᧦ઙߪฦゲᣇะ ߩቯᩰ⩄㊀ࠍන⁛ߢ⽶⩄ߔࠆ߽ߩߣߔࠆ㧚F_x^⽶⩄ᤨ

ߪ߭ߕߺࠥ࡯ࠫD3, B4, F_z^⽶⩄ᤨߪB2, M_z^⽶⩄ᤨ

ߪB3, B4ߩ߭ߕߺࠍߘࠇߙࠇ᳞߼ࠆ㧚

ᦨㆡൻߢߪฦゲᣇะ⩄㊀ߦኻᔕߔࠆ߭ߕߺߛߌߢ ߥߊ㧘ᐓᷤᚑಽߦߟ޿ߡ߽⠨ᘦߔࠆ. Ṷ▚⚿ᨐߩ৻

଀ߣߒߡ㧘⿠ᱡㇱߩ᪞ߦ߅ߌࠆ߭ߕߺߣゲᣇะᐳᮡ

ߩ㑐ଥࠍㄭૃߒߚ࿑ࠍ࿑5, ࿑6߅ࠃ߮࿑7ߦ␜ߔ㧚

ᑼ(3-1)ߢ f ࠍ⩄㊀ߩᦨᄢ୯ߢᱜⷙൻߔࠆߣ f ^ߪන

૏ⴕ೉ߣߥࠅ㧘İ C^{߇ᚑ┙ߔࠆ㧚࿑} 4 ߩࠃ߁ߦ࠮

ࡦࠨߩਛᔃࠍේὐᐳᮡߣߒ㧘ࠥ࡯ࠫ1, 2ߩ⾍ࠅઃߌ

૏⟎ᐳᮡࠍx₁, x₂ߣ߅ߊߣ㧘ฦ⩄㊀ߦኻᔕߔࠆ߭ߕ ߺߩᚑಽߪ x₂ ,

Fx

H x₂ ,

Fy

H x₁ , x₂

z

z M

F H

H ^ޓ ^ߩࠃ߁

ߦฦᐳᮡߩ㑐ᢙߣߒߡ⠨߃ࠄࠇ㧘ߘࠇࠄߪᐓᷤᚑಽ

(5)

߽฽߼ߡC਄ߢએਅߩࠃ߁ߦ⴫ߐࠇࠆ㧚

»»

»

¼ º

««

«

¬ ª

2 1

2 2

2 1

2 2

2 1

2 2

2 1

2 2

x x

z z

y x

z z

y x

z z

y x

z z

y x

M F

F F

M F

F F

M F

F F

M F

F F

H H

ޓ ޓ

ޓ

ޓ ޓ

ޓ

ޓ ޓ

ޓ

ޓ ޓ

ޓ

İ C ^(4-1)

એਅ㧘ᑼ(4-1)ߢฦᚑಽࠍᑼ(4-2)ߩࠃ߁ߦ⴫⃻ߔࠆ㧚

»»

¼ º

««

¬ ª

2 44 1 43 2 42 2 41

2 34 1 33 2 32 2 31

2 24 1 23 2 22 2 21

2 14 1 13 2 12 2 11

x x

H H

ޓ ޓ

ޓ

ޓ ޓ

ޓ

ޓ ޓ

ޓ

ޓ ޓ

ޓ

C ^(4-2)

ᑼ(4-2)ߢCߪᱜᣇⴕ೉ߥߩߢ․⇣୯ߪ࿕᦭୯ߣ৻

⥌ߔࠆ㧚ᧄ┨ߢߪCߩᦨᄢ࿕᦭୯ߣᦨዊ࿕᦭୯ߩᲧ ࠍ╙3┨ߢቯ⟵ߒߚ᧦ઙᢙߣߒ㧘⋡⊛㑐ᢙߦ↪޿ࠆ㧚

࿕᦭୯ࠍ᳞߼ࠆߣ㧘ᦨᄢ࿕᦭୯O_max^{߅ࠃ߮ᦨዊ࿕᦭}

୯O_min^{ߪએਅߩࠃ߁ߦߥࠆ㧚}

11 12 33 44

3 2 3

max

3

1 H H H H

O

P P Q P P Q

(4-3)

12 11

min H H

O (4-4) ߎߎߢ

^ ₁₁ ₁₂ ₃₃ ₄₄ ₃₄ ₄₃ ₃₃ ₄₄ ₁₃₃₁ ₁₄ ₄₁`

44 33 12 11 3 44 33 12 11

31 43 33 41 14 41 34 44 31 13 44 33 43 34 12 11

2 6 1 27

1 2 1

H H H H H H H H H H H H

H H H H H H H H

H H H H H H H H H H H H H H H H

u

P

(4-5)

^ `

11 12 33 44²

41 14 31 13 44 33 43 34 44 33 12 11

9 1 3 2 1

H H H H

H H H H H H H H H H H H

Q (4-6)

ߣߔࠆ㧚᧦ઙᢙߩᦨㆡ୯ߪ1ߥߩߢ㧘O_max^ߣOmin^ߩ

Ყ߆ࠄ1ࠍᒁ޿ߚ߽ߩߩ⛘ኻ୯ࠍᦨዊൻߔࠆࠃ߁ߦ㧘

⋡⊛㑐ᢙ(

x₁,x₂

ࠍએਅߩࠃ߁ߦቯ⟵ߔࠆ㧚

,

1

min max 2

1

( O

x O

x (4-7)

ᧄ࠮ࡦࠨߪ߭ߕߺࠥ࡯ࠫࠍ᪞ߩ஥㕙ߦ⾍ࠅઃߌࠆ ߚ߼㧘߭ߕߺࠥ࡯ࠫ߇᪞ߩ㕙ౝ߆ࠄߪߺ಴ߐߕ㧘߆ ߟࡈࠖ࡟࠶࠻ㇱߦដ߆ࠄߥ޿ࠃ߁ߦ⸳⸘ኸᴺࠍ⠨ᘦ ߒ㧘⋡⊛㑐ᢙߩቯ⟵ၞߪએਅߩㅢࠅߣߔࠆ㧚

5 . 7 9

.

5 dx₁d (4-8) 7

. 7 7

.

5 dx₂d (4-9)

ߐࠄߦ㧘߭ߕߺࠥ࡯ࠫห჻߇ធ⸅ߒߥ޿ࠃ߁ߥ೙⚂

᧦ઙߣߒߡ

2 1

1x t

x (4-10) ߇ടࠊࠅ㧘ᦨ⚳⊛ߥ೙⚂᧦ઙߪᑼ(4-8)́(4-10)ߩࠃ߁ ߦߥࠆ㧚ߎࠇࠄߩ೙⚂᧦ઙਅߢᑼ(4-7)ߦᑼ(4-3), (4-4) ࠍઍ౉ߒ㧘C⸒⺆ߦࠃߞߡࡊࡠࠣ࡜ࡒࡦࠣߒߚࠕ࡞

ࠧ࡝࠭ࡓࠍ↪޿ߡᑼ(4-7)ࠍᦨዊൻߒߚᤨߩx₁,x₂^ࠍ

᳞߼ࠆߣx₁ 6.7[mm], x₂ 5.7[mm]ߣߥࠅ㧘ߎࠇ ࠄ߇ࠥ࡯ࠫ1߅ࠃ߮ࠥ࡯ࠫ2ߩ⾍ࠅઃߌ૏⟎ᐳᮡߣ

᳿ቯߐࠇࠆ㧚

Fig.4. Coordinates of the Strain Gauges.

Fig.5. Approximate Function of F_x F_y .

Fig.6. Approximate Function of F_z.

x

z x

1

x

2

y x

z x

1

x

2

y

Fx

H

Fy

H

Fz

H

Mz

H

50 100 150 200

0

6 6.5 7 7.5 8

5.5

Strain[ ]Strain[ ]

-50

x[m^[m

x^[m^[m]^[m]

x^[m x ^[m]^[m]

Fx

H

Fy

H

Fz

H

Mz

H

Fx

H

Fy

H

Fz

H

Mz

H

50 100 150 200

0

6 6.5 7 7.5 8

5.5

Strain[ ]Strain[ ]

-50

x[m^[m

x^[m^[m]^[m]

x^[m x ^[m]^[m]

Fx

H

Fy

H

Fz

H

Mz

H

50 100 150 200

0

6 6.5 7 7.5 8

5.5

Strain[ ]Strain[ ]

-50

x[m[m

x[mm]m]

x[m

x m]m]

Fx

H

Fy

H

Fz

H

Mz

H

Fx

H

Fy

H

Fz

H

Mz

H

50 100 150 200

0

6 6.5 7 7.5 8

5.5

Strain[ ]Strain[ ]

-50

x[m[m

x[mm]m]

x[m

x m]m]

(6)

Fig.7. Approximate Function of M_z.

㧚⭯ဳ ゲജⷡ࠮ࡦࠨߩᕈ⢻⹏ଔ

⸘᷹୯ߩ⸃ᨆߦࠃࠆᗵᐲ⹏ଔ

4 ゲജⷡ࠮ࡦࠨߩ᦭㒢ⷐ⚛ࡕ࠺࡞ࠍ↪޿ߚࠪࡒࡘ

࡟࡯࡚ࠪࡦߦࠃߞߡ᳿ቯߒߚ⾍ࠅઃߌ૏⟎ࠍㆡ↪ߒ

᦭㒢ⷐ⚛⸃ᨆࠍ߅ߎߥ޿㧘ฦ⩄㊀᧦ઙߦኻᔕߒߚ߭

ߕߺࠍ᳞߼ࠆ㧚ߘߩᓟ߭ߕߺࠦࡦࡊ࡜ࠗࠕࡦࠬⴕ೉

ࠍ↪޿ߚᗵᐲ⹏ଔࠍ߅ߎߥ߁㧚ᗵᐲ⹏ଔߦ૶↪ߒߚ

᦭㒢ⷐ⚛ࡕ࠺࡞ࠍ࿑8ߦ␜ߔ㧚⸃ᨆ᧦ઙߪ᦭㒢ⷐ⚛

ࡕ࠺࡞ߩ▵ὐᢙ 1361, ⷐ⚛ᢙ 4086 ߢ޽ࠆએᄖߩ୯ ߪ╙4┨ߣห᭽ߩ߽ߩߣߒߚ㧚⸃ᨆ⚿ᨐߩ৻଀ߣߒ ߡ㧘߭ߕߺࠦࡦࡊ࡜ࠗࠕࡦࠬⴕ೉C^{㧘߭ߕߺࠥ࡯ࠫ}

ᗵᐲ C_i ^{㧘ജⷡᗵᐲ}V_min^{߅ࠃ߮᧦ઙᢙ}cond.C^ࠍߘࠇ ߙࠇᑼ(5-1)́(5-4)ߦ␜ߔ㧚

6844 . 0 0383 . 0 0019 . 0 0310 . 0 -

0390 . 0 .9329 0 0169 . 0 - 0058 . 0

0569 . 0 0239 . 0 5626 . 0 0554 . 0 -

0569 . 0 0239 . 0 0554 . 0 - 5626 . 0

¸¸

¸

¹

·

¨¨

¨

©

§

ޓ ޓ

ޓ

C (5-1)

6862 . 0 , 9339 . 0 , 5687 0 , 5687

0 2 3 4

1 C C C

C . . (5-2)

5045 .

min 0

V (5-3) 8634

. 1

cond.C (5-4)

࿕᦭୯⸃ᨆߦࠃࠆേ․ᕈߩᬌ⸽

ታ᷹ᤨߦ߅޿ߡ࠮ࡦࠨߦടࠊࠆᄖജߪᤨޘೞޘᄌ ൻߔࠆߚ߼㧘౒ᝄ߇⊒↢ߒ಴ജ୯ߦ⺋Ꮕ߇↢ߓࠆᕟ ࠇ߇޽ࠆ㧚ᧄ▵ߢߪ࿕᦭୯⸃ᨆࠍ߅ߎߥ޿㧘1 ᰴࡕ

࡯࠼ߦ߅ߌࠆ࠮ࡦࠨߩ࿕᦭ᝄേᢙࠍ⺞ߴ㧘4 ゲജⷡ

࠮ࡦࠨߩേ․ᕈࠍ⹏ଔߔࠆ㧚╙ 5.1▵ߣหߓ᧦ઙਅ ߦ߅ߌࠆ⸃ᨆ⚿ᨐߩ৻଀ߣߒߡ㧘1 ᰴࡕ࡯࠼ߦ߅ߌ ࠆ࿕᦭ᝄേᢙߪ₁_.₄₇_u₁₀⁴_[Hz]ߣߥߞߚ㧚ࡠࡏ࠶࠻ߩ

ࠨࡦࡊ࡝ࡦࠣ๟ᵄᢙߪ1000[Hz]⒟ᐲߢ޽ࠆߚ߼㧘࠮

ࡦࠨࠍࡠࡏ࠶࠻ࡂࡦ࠼ߦⵝ⌕ߒߡ߽౒ᝄߩᓇ㗀ߥߊ ᄖജࠍ⸘᷹น⢻ߣ޿߃ࠆ㧚

․ᕈ⹜㛎ߦࠃࠆ᦭ലᕈߩᬌ⸽

㐿⊒ߒߚ4ゲജⷡ࠮ࡦࠨ߅ࠃ߮╙4┨ߢ᳿ቯߒߚ

߭ߕߺࠥ࡯ࠫߩ⾍ࠅઃߌ૏⟎ߩ᦭ലᕈࠍᬌ⸽ߔࠆߚ

߼, 4ゲജⷡ࠮ࡦࠨࠍ⹜૞ߒ㧘ࠪࡒࡘ࡟࡯࡚ࠪࡦߦࠃ ߞߡ᳿ቯߒߚ⾍ࠅઃߌ૏⟎ࠍㆡ↪ߒ㧘ฦ⩄㊀ߦߟ޿

ߡએਅߩࠃ߁ߥᚻ㗅ߦࠃࠅ․ᕈ⹜㛎ࠍ߅ߎߥ߁㧚⵾

૞ߒߚ࠮ࡦࠨߩታᯏࡕ࠺࡞ࠍ࿑9ߦ␜ߔ㧚૶↪ߒߚ A/Dᄌ឵ེߪ౒๺㔚ᬺ⵾࠺࡯࠲ࡠࠟ࡯UCAM-60Bߢ

޽ࠆ㧚

y

x F

F, ^{ߦߟ޿ߡߪ㧘}F_x,F_x,F_y,F_y^ߩ⁴^ᣇะ ߦߟ޿ߡ߅߽ࠅࠍ 0.0[N]߆ࠄ98.0[N]߹ߢ 19.6[N]ߕ ߟ⽶⩄ߒ㧘ߘߩᓟ19.6[N]ߕߟ㒰෰ߒߡ⹜㛎ࠍ߅ߎߥ ߞߚ㧚Fzߦߟ޿ߡߪ㧘⹜㛎⠪ᚻ೨ᣇะࠍ

Fx

^ᣇะ ߣߒߚ႐วߣ㧘90[deg]࿁ォߒߡF_y^{ᣇะߣߒߚ႐ว} ߩ 2 ᣇะߦߟ޿ߡ߅߽ࠅࠍ 0.0[N]߆ࠄ 98.0[N]߹ߢ

19.6[N]ߕߟ⽶⩄ߒ㧘ߘߩᓟ19.6[N]ߕߟ㒰෰ߒߡ⹜㛎

ࠍ߅ߎߥߞߚ㧚

Mzߦߟ޿ߡߪ㧘࠮ࡦࠨਛᔃ߆ࠄ

┵߹ߢ 5[mm]㧘ਔ┵ߢ 10[mm]ߩࠕ࡯ࡓߦ߅߽ࠅࠍ

0.0[N]㨪9.8[N]߹ߢ1.96[N], 3.92[N], 9.8[N]ߣ޿߁ࠃ ߁ߦਔ┵หᤨߦタߖ㧘ߘߩᓟ㧘߅߽ࠅࠍਔ┵หᤨߦ 㒰෰ߒߡ⹜㛎ࠍ߅ߎߥߞߚ㧚ࠕ࡯ࡓߩਔ┵ߦ߅߽ࠅ ࠍษࠅਅߍࠆߚ߼㧘ᦨᄢߢ29.8[N]0.005[m]=

0.098[N࡮m]ߩࡕ࡯ࡔࡦ࠻߇⽶⩄ߐࠇࠆ㧚ߎߩᎿ⒟ࠍ

ࠕ࡯ࡓ߇x^ゲ߅ࠃ߮yゲߦᐔⴕߥ႐วߢߘࠇߙࠇ

Mz

, M_z^{ߦኻߒߡ߅ߎߥߞߚ㧚}

ᓧࠄࠇߚ⹜㛎⚿ᨐߩ৻଀ࠍ࿑10, ࿑11, ࿑12߅ࠃ

߮࿑13ߦ␜ߔ㧚ߎࠇࠄࠃࠅ㧘ฦゲᣇะߦኻߔࠆ಴ജ ߣߣ߽ߦ㧘ᐓᷤᚑಽߦߟ޿ߡ߽⩄㊀ߦኻߒߡ✢ᒻߩ

᜼േࠍ␜ߔߎߣ߇ಽ߆ࠆ㧚ࠃߞߡ㧘ᐓᷤ⵬ᱜⴕ೉ࠍ

↪޿ߡ࠮ࡦࠨߩ಴ജߦ฽߹ࠇࠆᐓᷤᚑಽࠍ⵬ᱜߒ߿

ߔ޿ߣ⠨߃ࠄࠇࠆ㧚߹ߚ㧘⹜㛎⚿ᨐ߆ࠄ᳞߼ߚ࠮ࡦ ࠨߩၮᧄ․ᕈ୯ߣߒߡ㧘ฦ⩄㊀ߦ߅ߌࠆ಴ജߩ㕖⋥

✢ᕈߣࡅࠬ࠹࡝ࠪࠬࠍ⴫2ߦ␜ߔ㧚ߎࠇࠄߪ⴫1ߩ

઀᭽⋡ᮡ୯ࠍḩߚߔߎߣ߇ಽ߆ࠆ㧚

ߐࠄߦ,⹜㛎⚿ᨐߩ৻଀ߣߒߡᗵᐲ⹏ଔࠍ߅ߎߥ ߁ߣ㧘߭ߕߺࠦࡦࡊ࡜ࠗࠕࡦࠬⴕ೉C^{㧘߭ߕߺࠥ࡯}

Fx

H

Fy

H

Fz

H

Mz

H

50 100 150 200

0

6 6.5 7 7.5 8

5.5

Strain[ ]Strain[ ]

-50

x [m^[m

x ^[m^[m]^[m]

x ^[m x ^[m]^[m]

Fx

H

Fy

H

Fz

H

Mz

H

Fx

H

Fy

H

Fz

H

Mz

H

50 100 150 200

0

6 6.5 7 7.5 8

5.5

Strain[ ]Strain[ ]

-50

x [m^[m

x ^[m^[m]^[m]

x ^[m x ^[m]^[m]

(7)

ࠫᗵᐲ C_i ^{㧘ജⷡᗵᐲ}V_min^{߅ࠃ߮᧦ઙᢙ}cond.C^ߪᰴ

ߩࠃ߁ߦᓧࠄࠇߚ㧚

6693 . 0 0642 . 0 - 0083 . 0 0083 . 0

0102 . 0 .0000 1 0008 . 0 - 0008 . 0 -

0215 . 0 - 0256 . 0 - 5393 . 0 0073 . 0

0037 . 0 - 0165 . 0 0073 . 0 5393 . 0

¸¸

¸

¹

·

¨¨

¨

©

§

ޓ ޓ

ޓ

ޓ ޓ

C (5-5)

6725 . 0 , 0001 . 1 , 5404 . 0 , 5396

0 2 3 4

1 C C C

C . (5-6)

5313 .

min 0

V (5-7) 8891

. 1

cond.C (5-8)

ᐓᷤ⵬ᱜ

࠮ࡦࠨߦ4ಽജ߇૞↪ߒߚ႐วߦ߅޿ߡ㧘⋧੕ᐓ

ᷤߦࠃࠅ↢ߓࠆ಴ജ୯ߪ૞↪ജߦኻߒߡ߶߷✢ᒻߥ 㑐ଥ߇޽ࠆߩߢ㧘⋧੕ᐓᷤߦࠃࠅ↢ߓࠆ಴ജ୯ࠍ੍

᷹ߔࠆߎߣ߇น⢻ߢ޽ࠆ㧚ࠃߞߡ㧘ᑼ(5-9)ߩ㑐ଥ߇ ᚑࠅ┙ߟ㧚㧚ߎߎߢ㧘

Fx, F_y, F_z, M_z^{ߪ⋧੕ᐓᷤ⵬}

ᱜ೨ߩ಴ജ୯㧘F_x, F_y^, Fz, M_z^{ߪ⋧੕ᐓᷤ⵬ᱜᓟ} ߩ಴ജ୯ߢ޽ࠅ㧘a₁₁,a₁₂,a₁₃,a₁₄,a₂₁,a₂₂,a₂₃,a₂₄,a₃₁,

44 43 42 41 34 33

32,a ,a ,a ,a ,a ,a

a ߪฦ⸘᷹୯ߦ߅ߌࠆㄭૃ

⋥✢ߩ௑߈ࠍ⴫ߔ㧚

z z

y x z

z z

y x z

z z

y x y

z z

y x x

M a F a F a F a M

M a F a F a F a F

44 43 42 41

34 33 32 31

24 23 22 21

14 13 12 11

(5-9)

ᑼ(5-9)ࠍⴕ೉ᒻᑼߢᦠߊߣ㧘

AF F

¸¸

¸

¹

·

¨¨

¨

©

§

¸¸

¸

¹

·

¨¨

¨

©

§

¸¸

¸

¹

·

¨¨

¨

©

§

z z y x

M F F F

a a a a

M F F F

44 43 42 41

34 33 32 31

24 23 22 21

14 13 12 11

(5-10)

ߎߎߢ㧘߭ߕߺ಴ജࠍH^ߣߔࠆߣ㧘F^ߣH^ߪ✢ᒻ㑐 ଥߦ޽ࠆߩߢ㧘Ყ଀ቯᢙࠍߘࠇߙࠇb₁,b₂,b₃,b₄ߣߒ㧘 ଥᢙⴕ೉Bࠍ

¸¸

¸

¹

·

¨¨

¨

©

§

4 3 2 1

0 0 0

0 0

0

0 0 0

b b b b

B (5-11)

ߣ߅޿ߡ

Bİ F

¸¸

¸

¹

·

¨¨

¨

©

§

¸¸

¸

¹

·

¨¨

¨

©

§

¸¸

¸

¹

·

¨¨

¨

©

§

z z y x

M F F F

z z y x

b b b b

M F F F

H H H H

4 3 2 1

0 0 0

0 0

0

0 0 0

(5-12)

ࠃࠅ㧘

¸¸

¸

¹

·

¨¨

¨

©

§

c c c c

¸¸

¸

¹

·

¨¨

¨

©

§

¸¸

¸

¹

·

¨¨

¨

©

§

44 43 42 41

34 33 32 31

24 23 22 21

14 13 12 11

4 3 2 1

44 43 42 41

34 33 32 31

24 23 22 21

14 13 12 11

0 0 0

a a a a

b b b b

a a a a

a a a

a (5-13)

ߣ߅޿ߡ㧘ᑼ(5-14)߇ዉ߆ࠇࠆ㧚

İ A

F c

¸¸

¸

¹

·

¨¨

¨

©

§

¸¸

¸

¹

·

¨¨

¨

©

§

c c c c

c c c

c

c c c c

¸¸

¸

¹

·

¨¨

¨

©

§

z z y x

M F F F

z z y x

a a a a

M F F F

H H H H

44 43 42 41

34 33 32 31

24 23 22 21

14 13 12 11

(5-14)

ߚߛߒ㧘Fߪ⵬ᱜᓟߩ⩄㊀୯㧘Acߪᄌ឵ⴕ೉㧘_İߪ

߭ߕߺ಴ജߢ޽ࠆ㧚ߎߎߢ㧘İ^ߣFߦኻߒߡฦ૞↪

ജࠍᄌ߃ߚታ㛎ࠍN^{࿁ⴕ޿㧘ᑼ}(5-15)ߦ␜ߔ⋧੕ᐓ

ᷤ⵬ᱜ೨ߩ⸘᷹୯ⴕ೉Ǽ㧘ᑼ(5-16)ߦ␜ߔ⋧੕ᐓᷤ

⵬ᱜᓟߩ⸘᷹୯ⴕ೉Fcࠍ᳞߼ࠆ㧚

İ İ_N

E 1 (5-15)

F FN

Fc 1 (5-16) ߎߎߢ㧘ᑼ(5-9)ߩ᜛ᒛ♽ࠍ⠨߃ࠆߣ㧘ᑼ(5-15)㧘(5-16) ࠃࠅᑼ(5-17)߇ᓧࠄࠇࠆ㧚

E A

Fc c (5-17)

ᑼ(5-17)ߦE^{ߩ৻⥸ൻㅒⴕ೉}E^{ࠍฝ߆ࠄਸ਼ߕࠆߎ}

ߣߦࠃࠅᑼ(5-18)߇ᓧࠄࠇࠆ㧚 c

c FE

A (5-18) ᑼ(5-18)ࠃࠅ㧘⋧੕ᐓᷤ⵬ᱜ೨ߩ⸘᷹୯߆ࠄ⋧੕ᐓᷤ

⵬ᱜᓟߩ⸘᷹୯ࠍ᳞߼ࠆᄌ឵ⴕ೉Ac^{߇▚಴ߢ߈ࠆ㧚}

࠮ࡦࠨߦฦ૞↪ജࠍ⽶⩄࡮㒰෰ߒ㧘ߘߩᤨߩ಴ജࠃ

ࠅᑼ(5-18)ߦ␜ߔᄌ឵ⴕ೉Ac^{ࠍ᳞߼㧘⋧੕ᐓᷤ⵬ᱜ}

ࠍⴕ߁ߎߣ߇น⢻ߣߥࠆ㧚╙ 5.3 ▵ߢ߅ߎߥߞߚ․

ᕈ⹜㛎ߩ⚿ᨐ㧘Acߪᑼ(5-19)ߩࠃ߁ߦ᳞߼ࠄࠇߚ㧚

¸¸

¸

¹

·

¨¨

¨

©

§

c

0.00284 0.00010

0.00003 0.00031

0.00188 0.07764

0.00191 0.00071

0.00041 0.00008

0.07217 0.00077

0.00219 0.00355

0.00425 05316

. 0

A (5-19)

(8)

ࡕ࡯ࡔࡦ࠻ߩᓇ㗀

ᧄ࠮ࡦࠨߪF_x, F_y, F_z, M_z^ߩ4ಽജࠍ⸘᷹ߔࠆ ߚ߼㧘x^ゲ߅ࠃ߮y^{ゲ࿁ࠅߩࡕ࡯ࡔࡦ࠻}M_x,M_y^ߩ ᓇ㗀ߦߟ޿ߡߪ⠨ᘦߒߡ߅ࠄߕ㧘ߎࠇࠄߩᐓᷤᚑಽ ߇↢ߓߚ႐ว㧘ᑼ(5-19)ߦ␜ߔᐓᷤ⵬ᱜⴕ೉ߢߪᓇ㗀 ࠍ⵬ᱜߢ߈ߥ޿㧚ߘߎߢ㧘᦭㒢ⷐ⚛⸃ᨆߦࠃߞߡࡕ

࠺࡞ߦM_z^{ߩቯᩰ⩄㊀ߣ╬ߒ޿}M_x,M_y= 0.1[N㨯m]

ߩࡕ࡯ࡔࡦ࠻ࠍ⽶⩄ߒ㧘߭ߕߺࠥ࡯ࠫߩ಴ജࠍ⺞ߴ ࠆ㧚⩄㊀᧦ઙߪ

y

x M

M ^߽ߒߊߪM_x,M_y^ߩ෺ᣇࠍ

⽶⩄ߔࠆ߽ߩߣߒ㧘╙ 5.1▵ߣหߓ᧦ઙਅߢ⸃ᨆࠍ ߅ߎߥߞߚ⚿ᨐߩ৻଀ߣߒߡ㧘߭ߕߺߩᦨᄢ಴ജߪ

1.5610^-5ߣߥߞߚ㧚ߎࠇߪᧄ┨ߢ߅ߎߥߞߚ᦭㒢ⷐ

⚛⸃ᨆ߅ࠃ߮․ᕈ⹜㛎ߦ߅޿ߡᓧࠄࠇߚᦨᄢ಴ജߩ 6.610^-3୚⒟ᐲߢ޽ߞߚߚ߼㧘M_x,M_y^{ߩᓇ㗀ߪή} ⷞߢ߈ࠆᄢ߈ߐߣ޿߃ࠆ㧚

ᒝᐲ⹏ଔ

৻⥸⊛ߥ㊄ዻ᧚ᢱߢߪ㧘㒠ફὐߦ⥋ࠆ߹ߢߪᒢᕈ ᄌᒻࠍߔࠆߩߢ㧘߭ߕߺߩᒢᕈ㒢⇇

Hmax^{ߪ㧘㒠ફᔕ}

ജV_Y^{ߣ᧚ᢱߩ❑ᒢᕈଥᢙ}E^{ࠍࡈ࠶ࠢߩᴺೣߦㆡ↪}

ߔࠆߎߣߢᰴᑼࠃࠅ▚಴ߢ߈ࠆ㧚

E VY

H_max (5-20)

৻ᣇ㧘⿥ޘࠫࡘ࡜࡞ࡒࡦߥߤߩࠕ࡞ࡒ࠾࠙ࡓว㊄ߢ ߪ㧘᣿⏕ߥ㒠ફ⃻⽎߇⷗ࠄࠇߥ޿㧚ߎߩࠃ߁ߥ㊄ዻ ࠍ৻⥸↪ㅜߢ↪޿ࠆ႐วߦߪ㧘㒰⩄ߒߚᓟߩ᳗ਭ߭

ߕߺ߇0.2[%]ߣߥࠆᔕജࠍ0.2[%]⠴ജ

2 .

V0 (߹ߚߪ⠴

ജ)ߣ⒓ߒߡ㧘ታ↪਄㧘㒠ફᔕജߣห᭽ߥᗧ๧ࠍᜬߟ ᒝߐߩၮḰߣߒߡ↪޿ࠆߎߣߣߐࠇߡ޿ࠆ㧚 ߒ߆ߒߥ߇ࠄ㧘੹࿁⵾૞ߔࠆജⷡ࠮ࡦࠨߪ᭴ㅧㇱ

᧚(⿠ᱡㇱ)ߦ᳗ਭ߭ߕߺ߇ᱷࠆߎߣ߇⸵ߐࠇߥ޿ߚ

߼㧘਄⸥ߩࠃ߁ߥᛒ޿ߪᅷᒰߢߪߥߊ㧘ࠃࠅዊߐߥ 㗔ၞࠍᒢᕈၞߣߒߡቯ߼ࠆᔅⷐ߇޽ࠆ㧚ߘߎߢ㧘᣿

⏕ߥ㒠ફ⃻⽎߇⷗ࠄࠇࠆ᧚ᢱߩ㒠ફὐߩ߭ߕߺ߇

0.2㨪0.3[%]ߢ޽ࠆߎߣߦ⌕⋡ߒ㧘ߎߩ୯ߩᦨዊ୯ߢ

޽ࠆ0.2[%]߹ߢߩ߭ߕߺߩ▸࿐ࠍޟ᧚ᢱߦᱷࠆ᳗ਭ

߭ߕߺ߇ήⷞߢ߈ࠆᒢᕈၞޠߣ⸃㉼ߔࠆߎߣߣߒߚ㧚 ߔߥࠊߜ㧘H_max 2.0u10³( 0.2[%])^{ࠍ↪޿ࠆߎߣ} ߣߒߚ¹⁴⁾㧚

ᧄ▵ߢߪ࠮ࡦࠨߦ↢ߓࠆᦨᄢ߭ߕߺࠍH_l^ߣߒߚ

ᤨ㧘᦭㒢ⷐ⚛⸃ᨆߦࠃߞߡ Hmax

H_l ^{߇ᚑࠅ┙ߟߎߣ} ࠍᬌ⸽ߔࠆ㧚⩄㊀᧦ઙߪਛᄩㇱߦฦゲᣇะߩቯᩰ⩄

㊀ࠍหᤨߦ⽶⩄ߔࠆ߽ߩߣߒ㧘╙ 5.1 ▵ߣหߓ᧦ઙ ਅߢ⸃ᨆࠍ߅ߎߥߞߚ⚿ᨐߩ৻଀ߣߒߡ㧘H_l

10 3

93 .

1 u ^{ߣߥߞߚ㧚ߎࠇߪ}H_lH_max^{ࠍḩߚߒߡ޿ࠆ} ߚ߼㧘࠮ࡦࠨߪቢోߦᒢᕈၞߢᄌᒻߒߡ߅ࠅ㧘ฦゲ ᣇะߦ૞↪ߔࠆᄖജࠍ⸘᷹น⢻ߢ޽ࠆ㧚

⠨ኤ

⸃ᨆ߅ࠃ߮․ᕈ⹜㛎ߩฦᗵᐲ⹏ଔ⚿ᨐࠍᲧセߔࠆ

ߣ㧘ᑼ(5-1), (5-5)ࠃࠅ෺ᣇߣ߽߭ߕߺࠦࡦࡊ࡜ࠗࠕࡦ

ࠬⴕ೉ߪኻⷺ㗄ߣᲧセߒߡ㕖ኻⷺ㗄߇චಽߦዊߐߊ ߥߞߡ߅ࠅ㧘ኻⷺ㗄ߩ୯߽෺ᣇߢ߶߷╬ߒߊߥߞߡ

޿ࠆ㧚࠮ࡦࠨߩ಴ജߦ฽߹ࠇࠆᐓᷤᚑಽ߇ዊߐ޿ߎ ߣ߆ࠄ㧘࠮ࡦࠨߪ4ಽജࠍߘࠇߙࠇ⁛┙ߒߚᚑಽߣ ߒߡᬌ಴น⢻ߥߎߣ߇ಽ߆ࠆ㧚

߹ߚ㧘ᑼ(5-2), (5-6)ࠃࠅ෺ᣇߣ߽ߦ

Fz^ߦኻߔࠆ߭

ߕߺࠥ࡯ࠫᗵᐲߣᲧセߒߡF_x, F_y, M_z^ߦኻߔࠆ߭

ߕߺࠥ࡯ࠫᗵᐲ߇ૐߊ㧘․ߦ

y

x F

F, ^ߦߟ޿ߡߪ 0.5

୚⒟ᐲߣߥߞߡ޿ࠆߎߣ߇ಽ߆ࠆ㧚ߎࠇߪ㧘ߖࠎᢿ ജߢ޽ࠆFx, Fy^{߇ု⋥ജߢ޽ࠆ}Fz^{ߣᲧセߒߡ⽶⩄}

ߒߦߊ޿਄㧘F_x, F_y, M_zߦኻᔕߔࠆ߭ߕߺࠍࡂ࡯ࡈ ࡉ࡝࠶ࠫ࿁〝ߦࠃࠆ߭ߕߺࠥ࡯ࠫ4ᨎಽߩ಴ജ߆ࠄ ᬌ಴ߒߡ޿ࠆߚ߼㧘߭ߕߺࠥ࡯ࠫࠍ8ᨎ↪޿ߚࡈ࡞

ࡉ࡝࠶ࠫ࿁〝߆ࠄ߭ߕߺࠍᬌ಴ߔࠆF_z^ߣᲧセߒߡ

಴ജ߇ዊߐߊߥߞߚߚ߼ߢ޽ࠆߣ⠨߃ࠄࠇࠆ㧚 ജⷡᗵᐲ߅ࠃ߮᧦ઙᢙߦߟ޿ߡߪ㧘ᑼ(5-3), (5-4) ߅ࠃ߮ᑼ(5-7), (5-8)߆ࠄᦨㆡ୯ߢ޽ࠆ1 ߦㄭ޿ߣߪ

޿߃ߥ޿⚿ᨐࠍ␜ߒߚ㧚ߎࠇߪ㧘࠮ࡦࠨߩ઀᭽਄

Fz

ߦኻߔࠆ಴ജߩߺ߇ઁߣᲧセߒߡᄢ߈ߊߥࠆߚ߼ߣ

⠨߃ࠄࠇࠆ㧚ߒ߆ߒ,ߘࠇࠄߩ୯ߪ෺ᣇߢ߶߷╬ߒߊ ߥߞߡ߅ࠅ㧘ታᯏࡕ࠺࡞ߦኻߔࠆ᦭㒢ⷐ⚛ࡕ࠺࡞ߩ ᅷᒰᕈߪᬌ⸽ߢ߈ߚ㧚

ࠃߞߡ㧘ᧄ࠮ࡦࠨߪฦゲᣇะߩᗵᐲߦᏅ߇޽ࠆ߽

ߩߩ㧘4 ಽജࠍ⁛┙ߒߚᒻߢ⸘᷹น⢻ߢ޽ࠅ㧘⸃ᨆ ߅ࠃ߮․ᕈ⹜㛎ߩฦ⹏ଔ⚿ᨐ߆ࠄ⷗ߡ߽㧘࠮ࡦࠨߩ

᦭ലᕈߪᬌ⸽ߢ߈ߚ㧚