回転機械の軸振動解析モデルの高精度化 Improvement of

ᅇ㌿ᶵᲔࡢ㍈᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬
Improvement of analytical accuracy
of 1D finite element model
for rotor-dynamics
ᖺ ᭶
㛗Ụ ಙ㢧
1
┠ḟ
୺࡞グྕY
➨ ❶ ᗎㄽ
⥴ゝ
ᚑ᮶ࡢ◊✲
㍈┤ᚄኚ໬㒊ࡢ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ໬
ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ໬
ᮏ◊✲ࡢ┠ⓗ࡜ᴫせ
ཧ⪃ᩥ⊩
➨ ❶ ࣮ࣟࢱࡢ᭤ࡆ᣺ືゎᯒ࡜ࡡࡌࡾ᣺ືゎᯒࡢࡓࡵࡢ ḟඖᱱࡢ᭷㝈せ⣲ࣔࢹࣝ
⥴ゝ
᭤ࡆ᣺ືࡢゎᯒࣔࢹࣝ
᭤ࡆ᣺ືゎᯒࣔࢹࣝࡢ⮬⏤ᗘ
᭤ࡆ᣺ືゎᯒࣔࢹࣝࡢ㐠ື᪉⛬ᘧ
᭤ࡆ᣺ືゎᯒࣔࢹࣝ࡟࠾ࡅࡿᅇ㌿㍈ࡢࣔࢹࣝ໬
᭤ࡆ᣺ືゎᯒࣔࢹࣝ࡟࠾ࡅࡿࢹ࢕ࢫࢡࡢࣔࢹࣝ໬
࣮ࣟࢱ඲యࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝ
ࡡࡌࡾ᣺ືࡢゎᯒࣔࢹࣝ
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢ⮬⏤ᗘ
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢ㐠ື᪉⛬ᘧ
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝ࡟࠾ࡅࡿᅇ㌿㍈ࡢࣔࢹࣝ໬
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝ࡟࠾ࡅࡿࢹ࢕ࢫࢡࡢࣔࢹࣝ໬
࣮ࣟࢱ඲యࡢࡡࡌࡾ᣺ືゎᯒࣔࢹࣝ
i
⤖ゝ
ཧ⪃ᩥ⊩
➨ ❶ ㍈┤ᚄኚ໬㒊ࡢ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ໬
⥴ゝ
㐺⏝⠊ᅖࡢ᳨ウ
㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿぢ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝ
໬᪉ἲ
㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࡢ➼౯᭤ࡆ๛ᛶ
ᅇ㌿ࡤࡡࡢ┤ิ⤖ྜ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿぢ࠿ࡅࡢ㍈
ࡢ᭤ࡆ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝ
ᅇ㌿ࡤࡡࡢ୪ิ⤖ྜ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈
ࡢ᭤ࡆ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝ
᭤ࡆ๛ᛶࡢ➼౯㍈ᚄࡢ᳨ウ
ᥦ᱌᪉ἲࡢጇᙜᛶࡢ᳨ド
ḟඖ᭷㝈せ⣲ࣔࢹࣝࡢ᭤ࡆ๛ᛶࡢ⿵ṇ
᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘ
㟼Ⲵ㔜࡟ࡼࡿ᭤ࡆኚᙧࡢホ౯
ᐇ㦂࡟ࡼࡿ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡢホ౯
ᐇᶵ┦ᙜࡢゎᯒࣔࢹࣝ࡟ࡼࡿ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡢホ౯
⤖ゝ
ཧ⪃ᩥ⊩
➨ ❶ ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࡢ᳨ウ
⥴ゝ
ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࡢホ౯
᥋ゐゎᯒ࡟ࡼࡿᔐࡵྜ࠸ᙉࡉ࡜㍈ࡢ᭤ࡆ๛ᛶࡢ㛵ಀࡢ᳨ウ
ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࡢᐇ㦂࡟ࡼࡿホ౯
ii
⤖ゝ
➨ ❶ ㍈┤ᚄኚ໬㒊ࡢࡡࡌࡾ๛ᛶࡢ㧗⢭ᗘ໬
⥴ゝ
㟼ⓗࢺࣝࢡ࡟ࡼࡿእ࿘ᅛᐃ෇ᰕࡢᙎᛶኚᙧ
㐺⏝⠊ᅖࡢ᳨ウ
㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹ
ࣝ໬᪉ἲ
㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࡢ➼౯ࡡࡌࡾ๛ᛶ
ᅇ㌿ࡤࡡࡢ┤ิ⤖ྜ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈
ࡢࡡࡌࡾ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝ
ᅇ㌿ࡤࡡࡢ୪ิ⤖ྜ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈
ࡢࡡࡌࡾ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝ
ࡡࡌࡾ๛ᛶࡢ➼౯㍈ᚄࡢ᳨ウ
ᥦ᱌᪉ἲࡢጇᙜᛶࡢ᳨ド
ḟඖ᭷㝈せ⣲ࣔࢹࣝࡢࡡࡌࡾ๛ᛶࡢ⿵ṇ
ࡡࡌࡾ࣮ࣔࢻࡢᅛ᭷᣺ືᩘ
㟼ⓗࢺࣝࢡ࡟ࡼࡿࡡࡌࡾゅࡢホ౯
ᐇᶵ┦ᙜࡢゎᯒࣔࢹࣝ࡟ࡼࡿࡡࡌࡾ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡢホ౯
⤖ゝ
ཧ⪃ᩥ⊩
➨ ❶ ᣺ືヨ㦂⤖ᯝ࡟ᇶ࡙ࡃ㍈᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬
⥴ゝ
≉ᛶ⾜ิࡢྠᐃࡢᴫせ
≉ᛶ⾜ิࡢྠᐃ᪉ἲ
ᮍ▱ࣃ࣓࣮ࣛࢱࡢྠᐃᡭ㡰
iii
๛య≉ᛶ࡜ᙎᛶ≉ᛶ࡟㛵ࡍࡿᶍᨃࡢᐇ ್࡟ࡼࡿㄗᕪࢆᣢࡘゎᯒࣔࢹࣝࡢ㧗⢭ᗘ
໬
᣺ືゎᯒࣔࢹࣝ
≉ᛶ⾜ิࡢྠᐃ
⤖ゝ
ཧ⪃ᩥ⊩
➨ ❶ ⥲ᣓ
ᮏ◊✲࡟ࡼࡾᚓࡽࢀࡓᡂᯝ
௒ᚋࡢᒎᮃ
ㅰ㎡
㛵㐃ሗ࿌
௜㘓
$ ㍈せ⣲ࡢ㉁㔞⾜ิ㸪๛ᛶ⾜ิ㸪ࢪࣕ࢖ࣟ⾜ิ
$ ᩿㠃୍ᵝᱱࡢ᭤ࡆ᣺ືࡢᅛ᭷᣺ືᩘ
ཧ⪃ᩥ⊩
$ ㍈ࡢ〇సᅗ
$ ࢱ࣮ࣅࣥ㸫Ⓨ㟁ᶵ࣮ࣟࢱࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝࡢㅖඖ
$ ࢱ࣮ࣅࣥ㸫Ⓨ㟁ᶵ࣮ࣟࢱࡢࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢㅖඖ
iv
୺࡞グྕ
A
㸸 ㍈ࡢ᩿㠃✚ (m2)
Din
㸸 ࢹ࢕ࢫࢡࡢෆᚄ (m)
Dout
㸸 ࢹ࢕ࢫࢡࡢእᚄ (m)
E
㸸 ࣖࣥࢢ⋡ (N/m2)
E'
㸸 ⿵ṇᚋࡢࣖࣥࢢ⋡ (N/m2)
G
㸸 ᶓᙎᛶಀᩘ (N/m2)
G'
㸸 ⿵ṇᚋࡢᶓᙎᛶಀᩘ (N/m2)
I
㸸 ㍈ࡢ᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ (m4)
I1
㸸 ኱ᚄ㍈ࡢ᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ (m4)
I1'
㸸 ㍈┤ᚄኚ໬㒊ࡢ኱ᚄ㍈ഃ᩿㠃ࡢ᭤ࡆ๛ᛶࡢ➼౯᩿㠃 2 ḟ
࣮࣓ࣔࣥࢺ (m4)
I2
㸸 ᑠᚄ㍈ࡢ᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ (m4)
I2'
㸸 ㍈┤ᚄኚ໬㒊ࡢᑠᚄ㍈ഃ᩿㠃ࡢ᭤ࡆ๛ᛶࡢ➼౯᩿㠃 2 ḟ
࣮࣓ࣔࣥࢺ (m4)
IA
㸸 ᩿㠃 A-A ࡟࠾ࡅࡿ᭤ࡆ๛ᛶࡢ➼౯᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ (m4)
Id
㸸 ㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿ᭤ࡆ๛ᛶࡢ➼౯᩿㠃 2 ḟ
࣮࣓ࣔࣥࢺ (m4)
Ieq 㸸 㛗ࡉ 9d/32 ࡢᑠᚄ㍈せ⣲ࡢ᭤ࡆ๛ᛶࡢ➼౯᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ
(m4)
Ip
㸸 ᅇ㌿㍈せ⣲ࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ (m4)
Ip1 㸸 ኱ᚄ㍈ࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ (m4)
Ip2 㸸 ᑠᚄ㍈ࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ (m4)
Ipeq 㸸㛗ࡉ 3ʌd/{64(1+0.3382ȕ3+0.0815ȕ5+0.1144ȕ6+0.0125ȕ7)}ࡢᑠᚄᱱ
せ⣲ࡢࡡࡌࡾ๛ᛶࡢ➼౯᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ (m4)
Ipd 㸸 ㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡡࡌࡾ๛ᛶࡢ➼౯᩿㠃 2 ḟ
ᴟ࣮࣓ࣔࣥࢺ (m4)
I''
JCG
㸸 ㍈┤ᚄኚ໬㒊᩿㠃ࡢ➼౯᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ (m4)
㸸 ࣮ࣟࢱࡢ㔜ᚰࢆ㏻ࡾᅇ㌿㍈࡟┤஺ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ
࣮࣓ࣔࣥࢺ (kgm2)
Jdisc
㸸 ࢹ࢕ࢫࢡࡢ┤ᚄ㍈࿘ࡾࡢ័ᛶ࣮࣓ࣔࣥࢺ (kgm2)
JpCG
㸸 ࣮ࣟࢱࡢᅇ㌿㍈࿘ࡾࡢ୺័ᛶ࣮࣓ࣔࣥࢺ (kgm2)
Jpdisc 㸸 ࢹ࢕ࢫࢡࡢᅇ㌿㍈࿘ࡾࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ (kgm2)
M
㸸 ᭤ࡆ࣮࣓ࣔࣥࢺ (N‚m)
T
㸸 ࢺࣝࢡ (N‚m)
v
a
㸸 ᑠᚄ㍈ࡲࡓࡣ๛య㍈ࡢ༙ᚄ (m)㸪a=d/2
b
㸸 ෇ᯈࡲࡓࡣ኱ᚄ㍈ࡢ༙ᚄ (m)㸪b=d0/2
c1㸪c2㸪c3 㸸 ࡓࢃࡳ᭤⥺ࡢ 3 ḟከ㡯ᘧ㏆ఝ᭤⥺ࡢಀᩘ(ษ∦ࢆ 0 ࡜ࡋࡓሙྜ)
d
㸸 ᑠᚄ㍈ࡢ㍈ᚄ (m)
d0
㸸 ෇ᯈࡲࡓࡣ኱ᚄ㍈ࡢ㍈ᚄ (m)
de
㸸 ➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿ኱ᚄ㒊ࡢ㍈ᚄ (m)
deq
㸸 ➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿᑠᚄ㒊ࡢ㍈ᚄ (m)
dpeq
㸸 ➼౯࡞ࡡࡌࡾ๛ᛶࢆ୚࠼ࡿᑠᚄ㒊ࡢ㍈ᚄ (m)
ds
㸸 ࢫ࣮ࣜࣈࡢෆᚄ (m)
f
㸸 ᅛ᭷᣺ືᩘ (Hz)
h
㸸 ෇ᯈࡲࡓࡣ኱ᚄ㒊ࡢཌࡉࡲࡓࡣ㍈᪉ྥ㛗ࡉ (m)
k1
㸸 ኱ᚄ㍈せ⣲ࡢᅇ㌿ࡤࡡᐃᩘ (N‚m/rad)
k2
㸸 ᑠᚄ㍈せ⣲ࡢᅇ㌿ࡤࡡᐃᩘ (N‚m/rad)
kd
㸸 㛗ࡉ 9d/32 ࡢせ⣲ࡢᅇ㌿ࡤࡡᐃᩘ (N‚m/rad)
keq 㸸 ➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿ㛗ࡉ 9d/32 ࡢせ⣲ࡢᅇ㌿ࡤࡡᐃᩘ
(N‚m/rad)
le
㸸 ㍈せ⣲ࡢ㛗ࡉ (m)
l2
㸸 ᑠᚄ㍈せ⣲ࡢ㛗ࡉ (m)
m1
㸸 ኱ᚄ㍈せ⣲ࡢ័ᛶ࣮࣓ࣔࣥࢺ (kg‚m2)
m2
㸸 ᑠᚄ㍈せ⣲ࡢ័ᛶ࣮࣓ࣔࣥࢺ (kg‚m2)
md
㸸 㛗ࡉ 9d/32 ࡢせ⣲ࡢ័ᛶ࣮࣓ࣔࣥࢺ (kg‚m2)
mdisc
㸸 ࢹ࢕ࢫࢡࡢ㉁㔞 (kg)
mR
㸸 ࣮ࣟࢱࡢ඲㉁㔞 (kg)
n
ndisc
㸸 ⠇Ⅼᩘ
㸸 ࢹ࢕ࢫࢡࡢᩘ
nf1
㸸 ᅛ᭷᣺ືᩘࡢᩘ
nf2
㸸 ᅛ᭷࣮ࣔࢻࡢᩘ
p
㸸 㠃ᅽⲴ㔜 (N)
p0
㸸 ᔐࡵྜ࠸࡟ࡼࡿ㠃ᅽⲴ㔜 (N)
tdisc
㸸 ࢹ࢕ࢫࢡࡢཌࡉ (m)
zCG
㸸 ㍈᪉ྥࡢ㔜ᚰ఩⨨ (m)
ǻd
㸸 ⥾ࡵ௦ (m)
ǻp
㸸 ᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ㠃ᅽⲴ㔜ࡢኚ໬ศ (N)
ĭ
㸸 ᅛ᭷࣮ࣔࢻ
Į
㸸 ᭤ࡆ࣮࣓ࣔࣥࢺ࡜ഴࡁゅࡢẚ౛ಀᩘ
Įi
㸸 ័ᛶಀᩘ
vi
Įm
㸸 ㉁㔞ಀᩘ
ȕ
㸸 ㍈┤ᚄẚ
Ȗ
㸸 ẚ౛ῶ⾶ᐃᩘ
į
㸸 ᱱࡢࡓࢃࡳ (m)
H
㸸 ᅛ᭷ゅ᣺ືᩘࡢᖹᆒㄗᕪ
ș
㸸 ࢺࣝࢡ T ࡟ࡼࡿᑠᚄ㍈ࡲࡓࡣ๛య㍈ࡢᅇ㌿ゅ (rad)
ț
㸸 ࡏࢇ᩿ᐃᩘ
ȝ
㸸 ᦶ᧿ಀᩘ
Ȟ
㸸 ࣏࢔ࢯࣥẚ
ȡ
㸸 ᅇ㌿㍈ࡢᐦᗘ (kg/m3)
ȡdisc
㸸 ࢹ࢕ࢫࢡࡢᐦᗘ (kg/m3)
ı
㸸 ᶆ‽೫ᕪ
IJ
㸸 ࡏࢇ᩿ಀᩘ
ȥ
㸸 ᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ㍈┤ᚄኚ໬㒊᩿㠃ࡢഴࡁゅ (rad)
Ȧn
㸸 ᅛ᭷ゅ᣺ືᩘ (rad/s)
vii
➨ ❶ ᗎㄽ
⥴ゝ
⌧ᅾ㸪ᗈ⠊࡞ศ㔝࡛ከ✀ከᵝ࡞ᅇ㌿ᶵᲔࡀ㸪✀ࠎࡢ᣺ືၥ㢟ࢆඞ᭹ࡋ࡚㸪㔜せ࡞ᙺ๭ࢆ
ᢸࡗ࡚࠸ࡿ㸬౛࠼ࡤ㸪Ⓨ㟁ศ㔝࡛ࡣ㸪1960 ᖺ௦௨㝆㸪㧗ฟຊ໬ࡢࡓࡵ࡟⵨Ẽࢱ࣮ࣅࣥࡢ኱
ᆺ໬㸪㛗ࢫࣃࣥ໬㸪ከࢫࣃࣥ໬ࡀ㐍ࡳ㸪༴㝤㏿ᗘࡢపୗ࡜ᐃ᱁ᅇ㌿ᩘ௨ୗ࡟Ꮡᅾࡍࡿ༴㝤
㏿ᗘࡢᩘࡢቑ኱࡜࠸࠺ၥ㢟࡟ᑐᛂࡍࡿࡓࡵ࡟㍈ཷࡢ๛ᛶ࡜ῶ⾶ࢆᐇ⏝࡟㊊ࡿ⢭ᗘ࡛᥎ᐃࡍ
ࡿᢏ⾡ࡸ୙㔮ྜ࠸᣺ືࢆపῶࡍࡿᢏ⾡࡞࡝ࡀ㛤Ⓨࡉࢀ㸪ᅜෆ࡛ࡣ 2000 ᖺ๓ᚋ࡟ 1000MW ⣭
ࡢ⵨Ẽࢱ࣮ࣅࣥࡀ✌ാࢆ㛤ጞࡋ࡚࠸ࡿ㸬ࡲࡓ㸪▼Ἔ࣭ኳ↛࢞ࢫศ㔝࡛ࡣ㸪ὶయᅽ㏦⏝ࡢ㐲
ᚰᅽ⦰ᶵࡢ㧗ᅽ໬㸪㧗㏿໬ࡀ㐍ࡳ㸪ࢩ࣮ࣝ㒊࡛Ⓨ⏕ࡍࡿ୙Ᏻᐃ໬ຊࢆዴఱ࡟ᢚ࠼ࡿ࠿ࡀタ
ィୖࡢ㔜せㄢ㢟࡜࡞ࡗ࡚࠾ࡾ㸪1990 ᖺ௦࠿ࡽ〇ရ໬ࡀ㐍ࢇ࡛ࡁࡓ☢Ẽ㍈ཷࢆไᚚࡋ࡚୙Ᏻ
ᐃ໬ຊࢆᢚ࠼ࡿᢏ⾡ࡀ㛤Ⓨࡉࢀ㸪Ᏻᐃᛶྥୖࡢࡓࡵࡢ᭦࡞ࡿᢏ⾡㛤Ⓨࡀ⥅⥆ࡉࢀ࡚࠸ࡿ㸬
ࡇࢀࡽࡢ௚࡟ࡶ㸪ᅇ㌿ᶵᲔࡢ኱ᆺ໬㸪ᑠᆺ໬㸪㧗㏿໬࡞࡝ࡢせồ࡟క࠸㛤Ⓨẁ㝵࡛Ⓨ⏕ࡍ
ࡿᵝࠎ࡞᣺ື⌧㇟ࡢⓎ⏕࣓࢝ࢽࢬ࣒ࡀ◊✲ࡉࢀ㸪ࡑࡢከࡃࡀゎ᫂ࡉࢀ࡚ࡁࡓ㸬㍈⣔ࡢ᣺ື
࡟㛵ࡋ࡚ࡣ㸪୙㔮ྜ࠸ࡸ࣑ࢫ࢔ࣛ࢖࣓ࣥࢺ㸦㑣㡲⏣௚㸪1983㸪ᩪ⸨㸪ᯇᮏ㸪1985㸪⚄ྜྷ௚㸪
2009㸧࡟㉳ᅉࡍࡿ᣺ື㸪๛ᛶࡀ㠀ᑐ⛠࡞ᅇ㌿㍈ࡢ୙Ᏻᐃ᣺ື㸪ࡍ࡭ࡾ㍈ཷࡢື≉ᛶ㸦ᰁ㇂㸪
1984㸪ᇼ㸪ຍ⸨㸪1989㸧㸪࢜࢖ࣝ࣍࢖ࢵࣉ㸦ᒸᓮ㸪ᇼ㸪1955㸪⯪ᕝ㸪ከࠎⰋ㸪1964㸪⏣ᾆ㸪
⏣୰㸪2007㸧㸪ࢫࢳ࣮࣒࣮࣍࣡ࣝ㸪ࢡࣛࢵࢡ㍈ࡢ᣺ື㸦▼⏣௚㸪1992㸪஭ୖ௚㸪2010㸧㸪ࣟ
࣮ࢱᇶ♏ບ᣺ᛂ⟅㸦⏣ᮧ௚㸪1981㸪ྂụ㸪▼ཎ㸪1983㸪ᒸᮏ௚㸪1992㸧㸪࣮ࣟࢱࡢෆ㒊ῶ⾶
࡟ࡼࡿ⮬ບ᣺ື㸦㔝Ἴ㸪ᐑୗ㸪1980㸧㸪ࢩ࣮ࣝ࡟ࡼࡿ୙Ᏻᐃ᣺ື㸦ᒾና௚㸪1983㸪⏣௚㸪
1988㸧㸪ᾮయෆⶶ࣮ࣟࢱࡢ୙Ᏻᐃ᣺ື㸦ᩧ⸨㸪ᰁ㇂㸪1978㸧㸪⋢㍈ཷࡢ㠀⥺ᙧࡤࡡ≉ᛶ࡟㉳
ᅉࡍࡿ㠀⥺ᙧ᣺ື㸦ᒣᮏ௚㸪1976㸧㸪ࢫࢡ࢖ࢬࣇ࢕࣒ࣝࢲࣥࣃ㍈ཷࡢື≉ᛶ㸦ྂụ㸪▼ཎ㸪
1987㸪⚄ྜྷ௚㸪2004㸧㸪☢Ẽ㍈ཷ㸦ྜྷ⏣௚㸪1993㸪ஂỌ௚㸪1996㸪ᒾኑ௚㸪1998㸪▼⏣௚㸪
2003㸪㧗ᶫ௚㸪2003, ⚄ྜྷ௚㸪2004㸧࡞࡝ࡀ◊✲ࡉࢀ࡚ࡁࡓ㸬⩼᣺ື࡟㛵ࡋ࡚ࡣ㸪ࢱ࣮ࣅࣥ
ື⩼࡟ຍࢃࡿὶయບ᣺ຊ㸦▼ཎ௚㸪1990㸧ࡸ࣑ࢫࢳ࣮ࣗࣥ⩼ࡢ᣺ື㸦㔠Ꮚ௚㸪1992㸧࡞࡝
ࡀ◊✲ࡉࢀ࡚ࡁࡓ㸬ᅇ㌿㍈ࡢ᭤ࡆ࡜ࡡࡌࡾࡢ㐃ᡂ᣺ື㸦ᒾን௚㸪1983㸧㸪ᅇ㌿㍈ࡢࡡࡌࡾ࡜
⩼᭤ࡆࡢ㐃ᡂ᣺ື㸦㣤ᓥ௚㸪1991㸧
㸪ᅇ㌿㍈ࡢ᭤ࡆ࡜⩼ࡢ᭤ࡆࡢ㐃ᡂ᣺ື㸦ⴗཎ㸪1982㸧࡟
ࡘ࠸࡚ࡶ◊✲ࡀ࡞ࡉࢀ࡚࠸ࡿ㸬ࡇࢀࡽࡢ◊✲ࡢᡂᯝࡣ㸪ࡼࡾ㧗ᛶ⬟㸪㧗ຠ⋡࡛⎔ቃ㈇Ⲵࡢ
ప࠸㸪Ᏻ඲࡞ᅇ㌿ᶵᲔࡢ㛤Ⓨࡸタィ࡟཯ᫎࡉࢀ࡚࠸ࡿ㸬
ࡋ࠿ࡋ㸪㏆ᖺ࡟࠾࠸࡚ࡶ㸪ᅇ㌿㍈ࡢ᭤ࡆ᣺ືࡸࡡࡌࡾ᣺ືࡀ஦ᨾ࡟⧅ࡀࡗࡓ஦౛ࡸ㸪ࢱ
࣮࣎ᶵᲔࡢࣈ࣮ࣞࢻࡸ࢖ࣥ࣌ࣛࡢ᣺ືࡀၥ㢟࡜࡞ࡗࡓ஦౛ࡀሗ࿌ࡉࢀ࡚࠸ࡿ㸬୙ලྜࡢせ
ᅉ࡜ࡋ࡚ࡣ㸪ᐇᶵࡢ᣺ືண ࡟⏝࠸ࡿゎᯒࣔࢹࣝࡢ⢭ᗘ୙㊊㸪㍈ཷ≉ᛶࡢኚ໬㸪እຊࡢぢ
✚⢭ᗘࡢ୙㊊࡞࡝࡟ຍ࠼࡚㸪᪤Ꮡࡢゎᯒࣔࢹ࡛ࣝ⾲⌧࡛ࡁ࡞࠸㠀⥺ᙧ᣺ື࡞࡝ࡢሙྜࡶ࠶
ࡿ㸬
ᅇ㌿ᶵᲔࡢ᣺ືࡣᙉไ᣺ື࡜୙Ᏻᐃ᣺ື࡟኱ูࡉࢀ㸪ࡇࢀࡽࡢ᣺ື⌧㇟ࡣ㸪࣮ࣟࢱࡢ㉁
㔞㸪๛ᛶ㸪ῶ⾶⾜ิ࡛⾲ࡉࢀࡿ࣮ࣟࢱࡢゎᯒࣔࢹࣝ࡟ᨭᣢ㒊ࡢ㉁㔞㸪๛ᛶ㸪ῶ⾶ࡢྛ⾜ิ
ࢆ㔜ࡡྜࢃࡏࡓ඲య⣔ࡢゎᯒࣔࢹࣝ࡟㸪እຊ࣋ࢡࢺࣝࢆ୚࠼࡚⾲⌧ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬㏻
1
ᖖࡢᅇ㌿ᶵᲔ࡛ࡣ㸪࣮ࣟࢱࡢῶ⾶࡜ᨭᣢ㒊ࡢ㉁㔞ࡣ↓どࡋ࡚ࣔࢹࣝ໬ࡀ࡞ࡉࢀࡿ㸬
ᙉไ᣺ືࡣ࣮ࣟࢱࡢ௵ពࡢ㒊఩࡟స⏝ࡍࡿᙉไຊ࡟ࡼࡗ࡚ᘬࡁ㉳ࡇࡉࢀࡿ᣺ື࡛࠶ࡾ㸪
௦⾲ⓗ࡞ࡶࡢ࡜ࡋ࡚୙㔮ྜ࠸᣺ືࡀ࠶ࡿ㸬୙㔮ྜ࠸᣺ືࢆపῶࡍࡿࡓࡵ࡟⾜ࢃࢀࡿ㔮ྜࢃ
ࡏ࡟㛵ࡍࡿ◊✲ࡣྂࡃ࠿ࡽ⾜ࢃࢀ࡚࠸ࡿ㸦▮㘠, ⏣ᮧ, 1969, ୕㍯, 1971, ሷᖭ, ⸨ἑ, 1979㸪
ⓑᮌ௚, 1980, ᯇୗ௚, 1992㸧㸬࣮ࣟࢱࢆ๛య࡜ࡋ࡚ᢅ࠺ࡇ࡜ࡀ࡛ࡁࡿప㏿ࡢᅇ㌿ᶵᲔ࡛ࡣᙳ
㡪ಀᩘἲ࡟ࡼࡿ 2 㠃ࣂࣛࣥࢩࣥࢢ࡛ẚ㍑ⓗᐜ᫆࡟㔮ྜࢃࡏࡿࡇ࡜ࡀ࡛ࡁࡿ㸬࣮ࣟࢱࢆᙎᛶ
య࡜ࡋ࡚ᢅ࠺ᚲせࡀ࠶ࡿ㧗㏿ࡢᅇ㌿ᶵᲔ࡛ࡣ㸪ᅛ᭷࣮ࣔࢻࢆ⪃៖ࡋࡓ࣮ࣔࢻ෇ࣂࣛࣥࢩࣥ
ࢢἲ㸦ྜྷᮏ, ᯇୗ, 2004, ᯇୗ௚, 2009㸧㸪࣮ࣟࢱࡢ㍈᣺ືపῶࡔࡅ࡛࡞ࡃ㍈ཷ཯ຊࢆࡶపῶ
ࡍࡿ n+2 㠃ἲ㸦▮㘠, ⏣ᮧ, 1969㸧㸪๓ྥࡁ୙㔮ྜ࠸᣺ືᡂศࡢࡳ࡟ὀ┠ࡋࡓࣂࣛࣥࢩࣥࢢἲ
㸦ᯇୗ௚, 2012㸧㸪ࡉࡽ࡟㸪୙㔮ྜ࠸ಟṇ㠃ࡢไ⣙ࡀ࠶ࡿ୰࡛ṧ␃᣺ືࡢୖ㝈್ࡸಟṇ㔞ࡢ
ୖ㝈್ࢆ⪃៖ࡋࡓ᭱㐺ࣂࣛࣥࢩࣥࢢἲࡀ㛤Ⓨࡉࢀ࡚࠸ࡿ㸦⚄ྜྷ௚㸪1999㸧㸬ᅇ㌿ᶵᲔࡢ᣺ື
ゎᯒࣔࢹࣝ࡟࠾ࡅࡿᨭᣢ㒊ࡢ๛ᛶ࡜ῶ⾶ࡢ⢭ᗘࡣ㸪࣮ࣟࢱࡢࡶࡢ࡜ẚ࡭࡚ప࠸ሙྜࡀከ࠸
ࡓࡵ㸪㔮ྜࢃࡏ࡛౑⏝ࡍࡿᙳ㡪ಀᩘࡣ㸪ゎᯒࣔࢹࣝ࠿ࡽ᥎ᐃࡋࡓࡶࡢ࡛ࡣ࡞ࡃ㸪ᐇ ࡋࡓ
᣺ື್࠿ࡽồࡵࡿ᪉ࡀ⢭ᗘ㠃࡛᭷฼࡛࠶ࡿ㸦ᒾና㸪1980㸧
㸬
୙Ᏻᐃ᣺ືࡣ㸪࣮ࣟࢱ࡜ᨭᣢ㒊࠿ࡽ࡞ࡿ඲య⣔ࡢ」⣲ᅛ᭷್ࡢᐇ㒊ࡀṇ࡜࡞ࡿ㸪ࡍ࡞ࢃ
ࡕῶ⾶ࡀ㈇࡜࡞ࡿ᮲௳࡛Ⓨ⏕ࡍࡿ⮬ບ᣺ື࡛࠶ࡿ㸬౛࠼ࡤ㸪ࣈ࣮ࣞࢻࡀྲྀࡾ௜ࡅࡽࢀࡓᅇ
㌿㍈ࡀ㸪㌴ᐊ࡜࿧ࡤࢀࡿࢣ࣮ࢩࣥࢢ࡟ᅛᐃࡉࢀࡓࡍ࡭ࡾ㍈ཷ࡛ᨭᣢࡉࢀࡓᵓ㐀ࢆᣢࡘ⵨Ẽ
ࢱ࣮ࣅ࡛ࣥࡣ㸪࢜࢖ࣝ࣍࢖ࢵࣉࡸࢫࢳ࣮࣒࣮࣍࣡ࣝ࡞࡝ࡢ⮬ບ᣺ືࡀⓎ⏕ࡍࡿྍ⬟ᛶࡀ࠶
ࡿࡓࡵ㸪᝿ᐃࡉࢀࡿ㐠㌿᮲௳࡛ᅇ㌿㍈⣔ࡢῶ⾶ࡀ㈇࡜࡞ࡽ࡞࠸ࡼ࠺࡞㍈ཷ௙ᵝࡀỴᐃࡉࢀ
ࡿ㸬ࡍ࡭ࡾ㍈ཷ࡟ࡘ࠸࡚ࡣྂࡃ࠿ࡽ◊✲ࡀ࡞ࡉࢀ࡚࠾ࡾ㸪⌮ㄽ࡜ᐇ㦂࡛㸪┿෇㍈ཷ㸪ከ෇
ᘼ㍈ཷ㸪ࢸ࢕ࣝࢸ࢕ࣥࢢࣃࢵࢻ㍈ཷࡢ๛ᛶ࠾ࡼࡧῶ⾶≉ᛶࡀồࡵࡽࢀ࡚࠸ࡿ㸬㧗ᅽ㐲ᚰᅽ
⦰ᶵ࡛ࡣ㸪ࢩ࣮ࣝ㒊ࡢᨭᣢ≉ᛶࡀ୙Ᏻᐃഃ࡟స⏝ࡋ࡚⮬ບ᣺ືࢆᘬࡁ㉳ࡇࡍሙྜࡀ࠶ࡿࡓ
ࡵ㸪ࢩ࣮ࣝ㒊ࡢᨭᣢ≉ᛶࡢண ࡀᢏ⾡ⓗ࡞ㄢ㢟࡜࡞ࡗ࡚࠸ࡿ㸬ࡑࡢࡓࡵ㸪CFD ゎᯒ࡟ࡼࡿ
ࢩ࣮ࣝ㒊ࡢὶయຊࡢண ࡟㛵ࡍࡿ◊✲㸦すᔱ௚, 2014㸧ࡸ㸪☢Ẽ㍈ཷ࡟ࡼࡿࢩ࣮ࣝ㒊ࡢື≉
ᛶࢆྠᐃࡍࡿ◊✲㸦Zutavern and Childs, 2008㸧ࡀ࡞ࡉࢀ࡚࠸ࡿ㸬ᚋ⪅ࡢ◊✲࡛ࡣ࣮ࣟࢱ࡜㍈
ཷࡢࣔࢹࣝ໬⢭ᗘࡀಖドࡉࢀ࡚࠸ࡿࡇ࡜ࡀ๓ᥦ࡜࡞ࡿ㸬୙Ᏻᐃ᣺ືࡢ㜵Ṇ࡟ࡣ㸪Ᏻᐃ㝈⏺
࠿ࡽ༑ศవ⿱ࡢ࠶ࡿ≧ែ࡛㐠㌿ࡍࡿᚲせࡀ࠶ࡿࡓࡵ㸪⢭ᗘࡢ㧗࠸」⣲ᅛ᭷್ゎᯒࡀᚲ㡲࡛
࠶ࡿ㸬
ᅇ㌿ᶵᲔࡢ᣺ືࢆᐃ㔞ⓗ࡟ண ࡍࡿᢏ⾡࡟ࡘ࠸࡚ࡶྂࡃ࠿ࡽ◊✲ࡀ࡞ࡉࢀ࡚࠾ࡾ㸪ఏ㐩
࣐ࢺࣜࢵࢡࢫἲ㸦Pestel and Leckie, 1963, ⳥ᆅ, 1969, Ishida and Yamamoto, 2012㸧ࡸ᭷㝈せ⣲
ἲ㸦ᯇୗ௚㸪1984㸧ࡀ㐺⏝ࡉࢀࡓᐇ⏝ⓗ࡞㍈᣺ື࠾ࡼࡧࡡࡌࡾ᣺ືゎᯒࢯࣇࢺ࢙࢘࢔ࡀ㛤
Ⓨࡉࢀ㸪ᐇ㝿ࡢタィ⌧ሙ࡛⏝࠸ࡽࢀ࡚࠸ࡿ㸬᭷㝈せ⣲ἲࢆ⏝࠸ࡓᅇ㌿ᶵᲔࡢゎᯒ⎔ቃࡣ㸪
ᅇ㌿㍈ࢆ 1 ḟඖࡢᱱせ⣲࡛ࣔࢹࣝ໬ࡍࡿ 1D ࣔࢹࣝゎᯒ࡜ 3 ḟඖࡢࢯࣜࢵࢻせ⣲࡛ヲ⣽࡟ࣔ
ࢹࣝ໬ࡍࡿ 3D ࣔࢹࣝゎᯒ࡟኱ูࡉࢀࡿ㸬ᅇ㌿㍈ࡢ᣺ືᙧែࡣ㸪᭤ࡆ᣺ື㸦ᶓ᣺ື㸧㸪ᅇ㌿
㍈᪉ྥࡢ᣺ື㸦⦪᣺ື㸧
㸪ࡡࡌࡾ᣺ື㸪࠾ࡼࡧࡑࢀࡽࡀ㐃ᡂࡋࡓ᣺ື࡟ศ㢮ࡉࢀࡿ㸬ᑐ㇟࡜
ࡍࡿ᣺ືᙧែ࡟ⓗࢆ⤠ࡗࡓ 1D ࣔࢹࣝゎᯒ⎔ቃ࡛ࡣ㸪ゎᯒࣔࢹࣝ࡟ࡼࡗ࡚ᢅ࠺ࡇ࡜ࡀ࡛ࡁࡿ
2
ᙧែ࡟ไ⣙ࡀ
ࡀከ࠸ࡀ㸪ィ
ィ⟬㈇Ⲵࡀᑠ
ᑠࡉ࠸㸬୍᪉
᪉㸪3D ࣔࢹࣝ
ࣝゎᯒ⎔ቃ࡛
࡛ࡣ㸪ᢅ࠺ࡇ
ࡇ࡜ࡀ
᣺ືᙧ
࡛ࡁࡿ᣺ືᙧែࡢ
ࡢไ⣙ࡀᑡ࡞
࡞࠸ࡀ㸪⌧ᅾ
ᅾ࡛ࡶィ⟬㈇
㈇Ⲵࡀᑠࡉ࠸
࠸࡜ࡣゝ࠼࡞
࡞࠸㸬ᅇ㌿ᶵ
ᶵᲔࡢ
༴㝤㏿ᗘ࣐ࢵ
ࢵࣉࡸ࣮ࣟࢱ
ࢱᙧ≧ࡢ᭱㐺
㐺໬࡞࡝ࡢ」
」⣲ᅛ᭷್ゎ
ゎᯒࢆ⧞ࡾ㏉
㏉ࡋ⾜
タィ࡟ᚲ㡲ࡢ㸪༴
せࡢ࠶ࡿィ⟬
⟬㸦ෆᾏ௚, 2013㸧࡛ࡣ 㸪ィ⟬㈇Ⲵࡀ᫬㛫ࢥࢫࢺ࡟┤⤖ࡍࡿ
ࡿ㸬
࠺ᚲせ
᣺ື
ືၥ㢟ࡢண඙
඙ࢆᤊ࠼࡚኱
኱஦࡟⮳ࡿ๓
๓࡟ᑐ⟇㸪ฎ
ฎ⨨ࢆ⾜࠺ࡓ
ࡓࡵ࡟㸪ᅇ㌿
㌿ᶵᲔࡢ≧ែ
ែ┘ど
ࢆ⾜࠺ࡓࡵࡢィ ࡜デ᩿࡟㛵
㛵ࡍࡿᢏ⾡ࡶ
ࡶ◊✲㛤Ⓨࡉ
ࡉࢀ㸪ᐇ⏝ࡉࢀ࡚࠸ࡿ㸬ࡲ
ࡲࡓ㸪〇సㄗ
ㄗᕪ㸪
≉ᛶࡢㄗᕪ㸪
㸪ࣔࢹࣝ໬ㄗ
ㄗᕪ࡞࡝࡟ࡼ
ࡼࡗ࡚㸪ゎᯒ
ᯒࣔࢹࣝ࡟ࡼ
ࡼࡿ㍈᣺ືࡢ
ࡢண ⢭ᗘࡀ
ࡀపୗ
ᮦᩱ≉
ࡍࡿሙ
ሙྜࡀ࠶ࡿ㸬ࡑࡇ࡛㸪ᅇ
ᅇ㌿ᶵᲔࡢ㛤
㛤Ⓨẁ㝵࡛ࡣ㸪ゎᯒࣔࢹࣝ
ࣝࡢ⢭ᗘࢆ☜
☜ㄆࡍࡿࡓࡵ
ࡵ࡟㸪
ᐇᶵࡢ
ࡢ࣮ࣟࢱ࡟ࡘ
ࡘ࠸࡚ᐇ㦂ࣔ
࣮ࣔࢻゎᯒ 㸦Ewins, 1984, ຍᕝ, ▼ᕝ
ᕝ, 1987, 㛗
㛗ᯇ, 1993, Balmes,
1996,, Maia and Silva,
S
1997㸧ࢆ⾜ࡗ࡚᣺ ື≉ᛶ࡜ࡋ࡚
࡚ᅛ᭷᣺ືᩘ
ᩘ㸪ᅛ᭷࣮ࣔ
࣮ࢻ㸪࣮ࣔࢻ
ࢻῶ⾶
ẚࢆ᥎
᥎ᐃࡋ㸪ゎᯒ
ᯒࣔࢹࣝࡢ⢭
⢭ᗘࢆ☜ㄆࡍ
ࡍࡿసᴗࡀ⾜
⾜ࢃࢀ࡚࠸ࡿ
ࡿ㸬≉࡟㸪ᐇ
ᐇ㦂࣮ࣔࢻゎ
ゎᯒ࡟
ࡼࡿᅛ᭷᣺ືᩘࡢ
ࡢ᥎ᐃ⢭ᗘࡣ
ࡣ㧗࠸ࡇ࡜ࡀ
ࡀ▱ࡽࢀ࡚࠸
࠸ࡿ㸬ࡋ࠿ࡋ
ࡋ㸪ᐇ㦂࣮ࣔ
࣮ࢻゎᯒࡢ⤖
⤖ᯝࡣ
⤖ᯝ࡜ࡢẚ㍑
㍑ࡢࡳ࡟౑⏝
⏝ࡉࢀࡿሙྜ
ྜࡀከࡃ㸪ࣟ
࣮ࣟࢱࡢ᣺ື
ືゎᯒࣔࢹࣝ
ࣝࡢ㧗⢭ᗘ໬
໬࡟༑
ゎᯒ⤖
ศ࡟ά
ά⏝ࡉࢀ࡚࠸
࠸ࡿ࡜ࡣゝ࠼
࠼࡞࠸㸬
ᅇ㌿
㌿ᶵᲔࡣ㸪ࡇࢀ࠿ࡽࡶ✀
✀ࠎ࡞ᡤ࡛౑
౑⏝ࡉࢀ⥆ࡅ
ࡅࡿ㸬ࡑࡢࡓ
ࡓࡵ㸪᣺ືၥ
ၥ㢟ࢆ㉳ࡇࡋ
ࡋ㞴࠸
ᅇ㌿ᶵ
ᶵᲔࡢ㛤Ⓨ㸪
㸪タィࢆຠ⋡
⋡ⓗ࡟⾜࠺ࡇ
ࡇ࡜ࡀồࡵࡽ
ࡽࢀࡿ㸬ࡑࡇ
ࡇ࡛㸪ィ⟬㈇
㈇Ⲵࡀᑠࡉࡃ
ࡃ㧗⢭
ᗘ࡞㍈
㍈᣺ື㸦᭤ࡆ
ࡆ᣺ື࠾ࡼࡧ
ࡧࡡࡌࡾ᣺ື
ື㸧ࡢゎᯒᡭ
ᡭἲࢆ㛤Ⓨࡍ
ࡍࡿࡇ࡜࡜㸪 ᐇ㦂࣮ࣔࢻ
ࢻゎᯒ
⤖ᯝ࡟ᇶ࡙࠸࡚ࣟ
࣮ࣟࢱࡢ᣺ື
ືゎᯒࣔࢹࣝ
ࣝࢆ㧗⢭ᗘ໬
໬ࡍࡿ᪉ἲࢆ
ࢆ㛤Ⓨࡍࡿࡇ
ࡇ࡜ࡀㄢ㢟࡛
࡛࠶ࡿ㸬
ᚑ᮶ࡢ◊✲
✲
࣮ࣟ
࣮ࢱࡢ 1D ࣔࢹࣝ࡟࠾࠸
ࣔ
࠸࡚㸪᭤ࡆ᣺
᣺ືゎᯒࣔࢹ
ࢹࣝࡢ㧗⢭ᗘ
ᗘ໬࡟㛵ࡍࡿ ᪤Ꮡ◊✲࡜ࡋ࡚㸪
㍈┤ᚄ
ᚄኚ໬㒊᩿㠃
㠃ࡢᙎᛶኚᙧ
ᙧ㸦ᅗ 1.1 ཧ
ཧ↷㸧࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ
ࡢ᭤ࡆ๛ᛶప
పୗࢆ㸪኱ᚄ
ᚄ㒊ࡢ
᭷ຠ┤
┤ᚄࡢῶᑡ࡟
࡟ࡼࡗ࡚⾲⌧
⌧ࡍࡿ᪉ἲ 㸦ᑠᇼ㸪196
66㸧࡜㸪ᙎᛶ
ᛶኚᙧ┦ᙜࡢ
ࡢ᭤ࡆ๛ᛶࢆ
ࢆ㍈ࡢ
᭤ࡆ๛
๛ᛶ࡟┤ิ⤖
⤖ྜࡍࡿ᪉ἲ
ἲ㸦㧗⏣௚㸪 2009㸧ࡀᥦ
ᥦ᱌ࡉࢀ࡚࠸
࠸ࡿ㸬
Big diameter partt
Small diameter
paart
Bending
moment
m
Fig. 1.11
Deformatiion of steppedd cross-sectio
on surface due to bending moment
3
ࡓ㸪1D ࣔࢹ
ࢹࣝ࡟࠾࠸࡚
࡚㸪ᅇ㌿㍈ࡢ ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࡢ
ࡢ᭷ຠ┤ᚄࢆ
ࢆ㸪ᔐࡵྜ࠸
࠸㒊ࡢ
ࡲࡓ
㍈᪉ྥ㛗ࡉ࡜㍈እ
እᚄࢆ㊊ࡋྜ
ྜࢃࡏࡓ್࡜
࡜ࡍࡿ᪉ἲࡀ
ࡀᥦ᱌ࡉࢀ࡚
࡚࠸ࡿ㸦Lalaanne and Feerraris,
㸧㸬ࡋ࠿ࡋ㸪1D ࣔࢹࣝ࡟
࡟࠾࠸࡚㸪㍈ ┤ᚄኚ໬㒊᩿
᩿㠃ࡢᙎᛶኚ
ኚᙧ࡟ࡼࡿࡡ
ࡡࡌࡾ๛ᛶప
పୗࢆ
1998㸧
⪃៖ࡋ࡚ࡡࡌࡾ᣺
᣺ືゎᯒࣔࢹ
ࢹࣝࢆ㧗⢭ᗘ
ᗘࡍࡿ᪉ἲࡣ
ࡣ㸪᪤Ꮡ◊✲
✲࡛ࡣぢᙜࡓ
ࡓࡽ࡞࠸㸬
D ࣔࢹࣝ࡟ࡼ
ࡼࡿ᣺ືゎᯒ
ᯒࡢຠ⋡໬ࢆ ᅗࡿ᪉ἲ࡜ࡋ࡚㸪ࣈ࣮ࣞ
࣮ࢻࡢ࿘ᮇᛶ
ᛶ࡟╔┠ࡋ࡚
࡚✵㛫
3D
ࣇ࣮࢚ࣜ⣭ᩘࡢᴫ
ᴫᛕࢆ㐺⏝ࡋ
ࡋࡓࣔࢹࣝ⦰
⦰⣙ἲࡀᥦ᱌
᱌ࡉࢀ࡚࠸ࡿ
ࡿ㸦A. Sternnchuss, E. Balmes,
㸧㸬
2006㸧
㍈┤
┤ᚄኚ໬㒊ࡢ
ࡢ᭤ࡆ๛ᛶࡢ
ࡢ㧗⢭ᗘ໬࡜
࡜ᔐࡵྜ࠸㒊
㒊ࡢ᭤ࡆ๛ᛶ
ᛶࡢ㧗⢭ᗘ໬
໬࡟㛵ࡍࡿ᪤
᪤Ꮡ◊
✲ࡢᴫ
ᴫせࢆ௨ୗ࡟
࡟♧ࡍ㸬
㍈┤ᚄኚ
ኚ໬㒊ࡢ᭤ࡆ
ࡆ๛ᛶࡢ㧗⢭
⢭ᗘ໬
኱ᚄ㒊
㒊ࡢ᭷ຠ┤ᚄ
ᚄ࡛ࡳ࠿ࡅࡢ
ࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⾲⌧
⌧ࡍࡿ᪉ἲ
1D
D ࣔࢹ࡛ࣝ㸪ᅇ㌿㍈ࡢ㍈
㍈┤ᚄኚ໬㒊 ᩿㠃ࡢᙎᛶኚ
ኚᙧ࡟ࡼࡿࡳ
ࡳ࠿ࡅࡢ㍈ࡢ
ࡢ᭤ࡆ๛ᛶప
పୗࢆ
⪃៖ࡋ࡚᭤ࡆ᣺ື
ືゎᯒࣔࢹࣝ
ࣝࢆ㧗⢭ᗘ໬
໬ࡍࡿ᪉ἲ࡜
࡜ࡋ࡚㸪ᑠᇼ
ᇼࡣ㸪኱ᚄ㒊
㒊ࡢ᭤ࡆ๛ᛶ
ᛶࢆ㐺
ᙅࡵࡿ᪉ἲࢆ
ࢆᥦ᱌ࡋ࡚࠸
࠸ࡿ㸬ᑠᇼࡣ
ࡣ㸪ᅗ 1.2 ࡟♧ࡉࢀࡿ㍈࡟
࡟࠾࠸࡚㸪㍈
㍈┤ᚄኚ໬㒊
㒊ࡢ᭤
ษ࡟ᙅ
ࡆ๛ᛶ
ᛶࡢపୗࢆ㸪➼౯᭤ࡆ๛
๛ᛶࢆ୚࠼ࡿ
ࡿ኱ᚄ㒊ࡢ㍈
㍈ᚄ de ࢆ⏝࠸
࠸࡚⾲ࡋࡓ㸬 ᘧ㸦1.1㸧ࡣ
ࡣ㸪ᑠ
ᇼ࡟ࡼࡗ࡚ᥦ᱌ࡉ
ࡉࢀࡓ㸪ᖜᚄ
ᚄẚ h/d ࡢ 3 ḟᘧ࡟ࡼࡗ
ࡗ࡚ de ࢆ୚࠼
࠼ࡿᘧ࡛࠶ࡿ 㸬
ȭd
ȭd e
ȭd 0
h
Fig. 1.2
1
de
Modelin
ng of shaft wiith stepped crross-section using
u
Kobori Method
2
3
ª
§h·
§h·
§h· º
d «1 0.44080¨ ¸ 0.04515¨ ¸ 0.0099¨ ¸ »
©d ¹
©d ¹
© d ¹ ¼»
¬«
㸦1.1㸧
㒊ࡢ➼౯㍈ᚄ
ᚄ de ࢆ୚࠼ࡿᘧ㸦1.1㸧 ࡢጇᙜᛶࡣᐇ
ᐇ㦂࡛ホ౯ࡉ
ࡉࢀ࡚࠸ࡿ㸬 ࡋ࠿ࡋ㸪➼
➼౯㍈
኱ᚄ㒊
ᚄ de ࡣ h/d ࡢࡳࡢ
ࡢ㛵ᩘ࡛࠶ࡾ
ࡾ኱ᚄ㒊ࡢ㍈
㍈እᚄ d0 ࡀ⪃
⪃៖ࡉࢀ࡞࠸
࠸ࡓࡵ㸪➼౯
౯㍈ᚄ de ࡀ኱
኱ᚄ㒊
እᚄ d0 ࡼࡾ኱ࡁࡃ࡞ࡿ࡜࠸࠺୙㐺ษ࡞⤖
⤖ᯝࢆ୚࠼ࡿ
ࡿሙྜࡀ࠶ࡿ
ࡿ㸬ᅗ 1.3 ࡣ h/d ࡜ de/d ࡢ㛵ಀ
ࡢ
4
࠶ࡿ㸬౛࠼ࡤ
ࡤ㸪h/d ࡀ 1 ࡢሙྜ࡛ࡣ de ࡀ 1.443 d ࡜࡞ࡾ㸪d 0<1.443 d ࡛࠶ࢀ
࡛
ࢆ♧ࡋࡓࡶࡢ࡛࠶
ࡢ᪉ἲࢆ㐺⏝
⏝ࡍࡿࡇ࡜ࡀ
ࡀ࡛ࡁ࡞࠸㸬
ࡤࡇࡢ
Fig. 1.3 Equivalent ddiameter of sh
haft on bendin
ng stiffness
ᙎᛶኚ
ኚᙧ┦ᙜࡢ᭤
᭤ࡆ๛ᛶࢆᱱ
ᱱࡢ᭤ࡆ๛ᛶ
ᛶ࡟┤ิ⤖ྜ
ྜࡋ࡚ࡳ࠿ࡅ
ࡅࡢ㍈ࡢ᭤ࡆ
ࡆ๛ᛶ
పୗࢆ⾲⌧ࡍࡿ᪉
᪉ἲ
㍈┤
┤ᚄኚ໬㒊ࢆ
ࢆᣢࡘᙎᛶ㍈
㍈࡛ࡣ㸪ࡑࡢ
ࡢᙎᛶ࡟ࡼࡗ
ࡗ࡚㸪᭤ࡆࣔ
࣮࣓ࣔࣥࢺࡀ
ࡀస⏝ࡍࡿ࡜
࡜㸪ᑠ
ᚄ㍈➃
➃㒊࡛ࡢ୰❧
❧㍈ࡢ᪉ྥࡀ
ࡀ㸪኱ᚄ㍈ഃ
ഃ᩿㠃ࡢἲ⥺
⥺᪉ྥ࡟ᑐࡋ
ࡋ࡚ഴࡃ⌧㇟
㇟ࡀ㉳ࡇࡿ㸬ࡘࡲ
ࡾ㸪኱
኱ᚄ㍈ഃ᩿㠃
㠃࡟ᑐࡍࡿᑠ
ᑠᚄ㍈➃㒊ࡢ
ࡢᅛᐃ≧ែࡣ
ࡣ᏶඲ᅛᐃ࡛
࡛ࡣ࡞࠸࡜ゝ
ゝ࠼ࡿ㸬ࡇࡢ
ࡢ⌧㇟
ࡣ㍈┤
┤ᚄኚ໬㒊࡟
࡟࠾࠸࡚㍈ࡢ
ࡢࡳ࠿ࡅࡢ᭤
᭤ࡆ๛ᛶࡀప
పୗࡍࡿ⌧㇟
㇟࡜ࡋ࡚ᢅ࠺ ࡇ࡜ࡀ࡛ࡁࡿ㸬
1D
D ࣔࢹ࡛ࣝ㸪ᅇ㌿㍈ࡢ㍈
㍈┤ᚄኚ໬㒊 ᩿㠃ࡢᙎᛶኚ
ኚᙧ࡟ࡼࡿࡳ
ࡳ࠿ࡅࡢ㍈ࡢ
ࡢ᭤ࡆ๛ᛶప
పୗࢆ
⪃៖ࡋ࡚᭤ࡆ᣺ື
ືゎᯒࣔࢹࣝ
ࣝࢆ㧗⢭ᗘ໬
໬ࡍࡿ᪉ἲ࡜
࡜ࡋ࡚㸪㧗⏣
⏣ࡽࡣ㸪ᙎᛶ
ᛶኚᙧ࡟┦ᙜ
ᙜࡍࡿ
ࡤࡡࢆ┤ิ⤖
⤖ྜ࡟ࡼࡗ࡚
࡚⾲⌧ࡍࡿ᪉
᪉ἲࢆᥦ᱌ࡋ
ࡋ࡚࠸ࡿ㸬
ᅇ㌿ࡤ
ᅗ 1.4 ࡣ㸪኱ᚄ
ᚄ㍈እ࿘㠃ࡀ
ࡀᅛᐃࡉࢀࡓ
ࡓ㍈┤ᚄኚ໬㒊ࢆᣢࡘ㍈࡟࠾࠸࡚㸪ᑠ
ᑠᚄ㍈ࡢඛ➃
➃࡟᭤
࣮࣓ࣥࢺ M ࡀస⏝ࡋࡓሙ
ሙྜ࡟㍈┤ᚄ
ᚄኚ໬㒊᩿㠃
㠃 A-A ࡀᙎᛶ
ᛶኚᙧࡍࡿࡇ
ࡇ࡜࡟ࡼࡗ࡚
࡚᩿㠃
ࡆ࣮ࣔ
A-A ࡢἲ⥺࡟ゅᗘ
ᗘ ȥ ࡢഴࡁࡀ⏕ࡌࡿᵝᏊ
Ꮚࢆ♧ࡋࡓࡶ
ࡶࡢ࡛࠶ࡿ㸬࡞࠾㸪ᅗ 11.4 ࡣ MATL
LAB®
ࡢࢧ࣮
࣮ࢻࣃ࣮ࢸ࢕
࢕ Toolbox ࡛࠶ࡿ
࡛
SDToools ♫〇ࡢ Structural Dyn
namics Toolbbox ࢆ౑⏝ࡋ
ࡋ࡚స
ᡂࡋࡓ
ࡓ 3D ࣔࢹࣝ
࡛ࣝ࠶ࡿ㸬
5
y
A
h
l2
Elastiic
defformation
ȭd
ȥ
ȭd 0
x
z
M
Elasstic
deformation
Fixedd A
Figg. 1.4
Deformation of sttepped cross-ssection due to
o bending mooment
ࡁゅ ȥ ࡢ㛫
㛫࡟ࡣᘧ㸦1..2㸧ࡢ㛵ಀࡀ
ࡀᡂࡾ❧ࡘ 㸦 Timoshenk
ko and
᭤ ࡆ࣮࣓ࣔࣥࢺ M ࡜ഴࡁ
Woinnowsky-Kriegger, 1973㸧㸬
DEh 3\ 㸦1.2㸧
M
M 㸸᭤ࡆ࣮ࣔ
࣮࣓ࣥࢺ (N
Nm)
Į 㸸᭤ࡆ࣮ࣔ
࣮࣓ࣥࢺ࡜ഴ
ഴࡁゅࡢẚ౛
౛ಀᩘ
ȥ 㸸㍈┤ᚄኚ
ኚ໬㒊᩿㠃ࡢ
ࡢഴࡁゅ (rad)
㸸෇ᯈࡢཌ
ཌࡉ㸦኱ᚄ㍈
㍈ࡢ㍈᪉ྥ㛗
㛗ࡉ㸧 (m)
h
⋡ (N/m2)
E 㸸ࣖࣥࢢ⋡
ᘧ㸦1.2㸧࡟࠾ࡅࡿ Į ࡣ㸪෇
෇ᯈࡢཌࡉ h ࡜┤ᚄ d0 ࡜ࡢẚ
࡜
h/d0 ࡀࢮࣟ࡟㏆௜
ࡀ
௜ࡃ࡜ᘧ㸦1..3㸧࡛
ࢀ㸪h/d0 ࡀ኱
኱ࡁࡃ࡞ࡿ࡜
࡜ᘧ㸦1.4㸧࡛
࡛♧ࡉࢀࡿ㸬
♧ࡉࢀ
D
D
E 2 1
S
˜
3 1 Q 2 E 2 1 E 2 1 ln E
S § Ed 0 ·
˜¨
¸
18 ¨© h ¸¹
§ h
·
, ¨¨
# 0 ¸¸
© d0
¹
㸦1.3㸧
3
㸦1.4㸧
6
ᘧ㸦1.3㸧ࡣ㍈┤ᚄኚ໬㒊ࡢ┤ᚄẚ ȕ (=d/d0)࡜࣏࢔ࢯࣥẚ Ȟ ࡢࡳࡢ㛵ᩘ࡛⾲ࡉࢀ㸦Darlow, et
al., 1980㸧㸪ᘧ㸦1.4㸧ࡣ෇ᯈࡢཌࡉ h㸪኱ᚄ㍈┤ᚄ d0㸪㍈┤ᚄẚ ȕ ࡛⾲ࡉࢀࡿ㸦㧗⏣௚㸪2009㸪
ᯘ௚㸪1990㸧
㸬
㧗⏣ࡽࡣ㸪ᘧ㸦1.4㸧࡜㛗ࡉ l ࡢ∦ᣢࡕᱱࡢ⌮ㄽᘧࡢ㛵ಀ࠿ࡽồࡵࡓ㍈┤ᚄኚ໬㒊᩿㠃ࡢ
ᙎᛶኚᙧศࡢ᭤ࡆ๛ᛶ EId ࡣᘧ㸦1.5㸧࡛⾲ࡉࢀ㸪㍈┤ᚄኚ໬㒊ࡢ➼౯࡞᭤ࡆ๛ᛶ EI’ࡣ EId
࡜㏻ᖖࡢᱱࡢ᭤ࡆ๛ᛶ EI ࢆ┤ิ⤖ྜࡋࡓᘧ㸦1.6㸧࡛⾲ࡉࢀࡿ࡜ࡋ࡚࠸ࡿ㸬
EI d
4
E Dh 3 l 㸦1.5㸧
9
EI c
EI ˜ EI d
㸦1.6)
EI EI d
ᘧ㸦1.3㸧ࡀᡂࡾ❧ࡘ h d 0 # 0 ࡢⷧ࠸෇ᯈࡢሙྜࡣ㸪෇ᯈࡀ㍈ࡢ᭤ࡆ๛ᛶ࡟ཬࡰࡍᙳ㡪ࢆ↓
ど࡛ࡁࡿ㸬ࡋ࠿ࡋ㸪ᘧ㸦1.4㸧 ࡀᡂࡾ❧ࡘ≧ἣ࡛ࡣ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿ᭤
ࡆ๛ᛶࡢపୗࡀ㉳ࡇࡿ㸬ࡑࡇ࡛㸪㧗⏣ࡽࡣ㸪ᘧ㸦1.4㸧 ࡀᡂࡾ❧ࡘሙྜࡢ㍈┤ᚄኚ໬㒊᩿㠃
ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ㸪ᘧ㸦1.6㸧ࡢᅇ㌿ࡤࡡࡢ┤ิ⤖ྜ࡛⾲⌧ࡋ
࡚࠸ࡿ㸬ࡲࡓ㸪㧗⏣ࡽࡣ㸪3 ḟඖ᭷㝈せ⣲ἲᩘ್ゎᯒࢥ࣮ࢻ SAMCEF-ROTOR ࢆ⏝࠸࡚㸪㍈
┤ᚄẚ ȕ=0.3175 ࡢሙྜ࡟ᘧ㸦1.4㸧 ࡀᡂࡾ❧ࡘ h/d0 ࡢ⠊ᅖࡀ h/d0•0.2 ࡛࠶ࡿࡇ࡜ࢆ♧ࡋ࡚
࠸ࡿ㸬
㧗⏣ࡽࡢ᪉ἲ࡛ࡣ㸪ᘧ㸦1.6㸧࡟♧ࡉࢀࡿࡼ࠺࡟㸪㛗ࡉࢆᣢࡓ࡞࠸ᅇ㌿ࡤࡡ EId ࢆ㍈せ⣲ࡢ
∦ഃࡢ⠇Ⅼࡢᅇ㌿ࡤࡡ࡟┤ิ⤖ྜࡍࡿᚲせࡀ࠶ࡿ㸬1 ḟඖᱱࡢ᭷㝈せ⣲ࣔࢹࣝࢆ⏝࠸ࡓ㍈᣺
ືゎᯒࢯࣇࢺ࢙࢘࢔࡛ࡣ㸪᭷㝈㛗ࡢ㍈せ⣲࡛ࡢࡳᅇ㌿ࡤࡡࡀᐃ⩏ࡉࢀ㸪㞄᥋ࡍࡿ㍈せ⣲ࡢ
ᅇ㌿ࡤࡡࡣ㔜」⠇Ⅼ࡟࠾࠸࡚㔜ࡡྜࢃࡏ࡟ࡼࡗ࡚⤖ྜ㸦୪ิ࡟⤖ྜ㸧ࡉࢀࡿࡓࡵ㸪≉ᐃࡢ
⠇Ⅼ࡛ᅇ㌿ࡤࡡࢆ┤ิ࡟⤖ྜࡍࡿࡇ࡜ࡀᅔ㞴࡜ᛮࢃࢀࡿ㸬
ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ໬
ᅗ 1.5 ࡣ㸪Lalanne ࡽ࡟ࡼࡗ࡚♧ࡉࢀ࡚࠸ࡿ㸪1D ࣔࢹࣝ࡟࠾ࡅࡿᔐࡵྜ࠸㒊ࡢ➼౯࡞᭤ࡆ
๛ᛶࢆ୚࠼ࡿ┤ᚄࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬(a)ࡣ㸯ࡘࡢࢹ࢕ࢫࢡせ⣲ࡀ 1 ࡘࡢ⠇Ⅼ࡟⤖ྜࡉࢀ
ࡓࣔࢹ࡛ࣝ࠶ࡾ㸪ࢹ࢕ࢫࢡせ⣲ࡀ᭤ࡆ๛ᛶ࡟ᙳ㡪ࢆ୚࠼࡞࠸ࣔࢹ࡛ࣝ࠶ࡿ㸬(b)ࡣ㍈ᚄ d ࡟
ࢹ࢕ࢫࢡᖜ h ࢆຍ࠼ࡓ d+h ࢆ➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿ㍈ᚄ࡜ࡋ࡚୰ኸࡢ⠇Ⅼ 2 ࡟ࢹ࢕ࢫࢡ
せ⣲ࡀ⤖ྜࡉࢀࡓࣔࢹ࡛ࣝ࠶ࡿ㸬(c)ࡣ(b)࡜ྠࡌࡃ➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿ㍈ᚄࢆ d+h ࡜ࡋ
࡚ 3 ࡘࡢ⠇Ⅼ 1, 2, 3 ࡢࡑࢀࡒࢀ࡟ᖜ h/4, h/2, h/4 ࡢࢹ࢕ࢫࢡせ⣲ࡀ⤖ྜࡉࢀ࡚࠸ࡿࣔࢹ࡛ࣝ
࠶ࡿ㸬
7
(a)
Fig. 1.5
(b)
(cc)
Conventional method of cconsidering sh
hrink fit effecct to the bendding stiffness
ࢹ࢕ࢫࢡࡸࢫ࣮ࣜࣈࡢᔐࡵ
ࡵྜ࠸࡟ࡼࡿ
ࡿ㍈ࡢ᭤ࡆ๛
๛ᛶࡢቑ኱ࡀ
ࡀ↓ど࡛ࡁࡿ
ࡿሙྜࡣᅗ 1.5
1 (a)
ࢹࣝ໬ࡀጇᙜ
ᙜ࡜⪃࠼ࡽࢀ
ࢀࡿ㸬୍᪉㸪 ᔐࡵྜ࠸࡟
࡟ࡼࡿ㍈ࡢ᭤
᭤ࡆ๛ᛶࡢቑ
ቑ኱ࡀ↓ど࡛
࡛ࡁ࡞
ࡢࣔࢹ
࠸ሙྜ
ྜࡣᅗ 1.5 (b)
( 㸪(c)ࡢ᪉ἲ࡛ࣔࢹࣝ໬
໬ࡍࢀࡤ㸪ྠ
ྠᅗ(a)ࡢࣔࢹ
ࢹࣝࡼࡾጇᙜ
ᙜ࡞᣺ື≉ᛶ
ᛶࢆᚓ
ࡿࡇ࡜ࡀ࡛ࡁࡿ࡜
࡜⪃࠼ࡽࢀࡿ
ࡿ㸬ࡋ࠿ࡋ㸪 ᅗ 1.5 (b)㸪(c)ࡢ᪉ἲ࡛
࡛ࡣᔐࡵྜ࠸
࠸࡟ࡼࡿ➼౯
౯㍈ᚄ
ศࡀࢹ࢕ࢫࢡ
ࢡࡸࢫ࣮ࣜࣈ
ࣈࡢᖜ h ࡢࡳ
ࡳ࡛୚࠼ࡽࢀ
ࢀࡿࡇ࡜ࡢ⌮
⌮ㄽⓗ࡞⫼ᬒ
ᬒࡀ୙࡛᫂࠶ࡿ㸬
ࡢቑศ
ᮏ◊✲ࡢ┠
┠ⓗ࡜ᴫせ
◊✲࡛ࡣ㸪ィ
ィ⟬㈇Ⲵࡢᑠ
ᑠࡉ࠸ 1 ḟ ඖᱱࡢ᭷㝈せ
せ⣲ἲࢆ㐺⏝
⏝ࡍࡿ㸪࣮ࣟ
࣮ࢱࡢ㍈᣺ື
ືゎᯒ
ᮏ◊
ࣔࢹࣝ
ࣝࢆ⏝࠸ࡓ㍈
㍈᣺ືゎᯒࡢ
ࡢ㧗⢭ᗘ໬ࢆ
ࢆᅗࡿࡓࡵ࡟
࡟㸪ḟࡢᅄࡘ
ࡘࡢࣔࢹࣝ໬
໬᪉ἲࢆ☜❧
❧ࡍࡿ
ࡇ࡜ࢆ┠ⓗ࡜ࡋ࡚
࡚࠸ࡿ㸬᭱ึ
ึࡢ୕ࡘࡣ᭤
᭤ࡆ᣺ືゎᯒ
ᯒ࡟㛵ࡍࡿࡶ
ࡶࡢ࡛࠶ࡾ㸪 ㍈┤ᚄኚ໬
໬㒊᩿
ᙎᛶኚᙧࢆ⪃
⪃៖ࡋࡓࣔࢹ
ࢹࣝ໬᪉ἲ㸪 ᔐࡵྜ࠸㒊
㒊ࡢࣔࢹࣝ໬
໬᪉ἲ㸪࠾ࡼ
ࡼࡧ᣺ືヨ㦂
㦂⤖ᯝ
㠃ࡢᙎ
࡟ᇶ࡙
࡙࠸࡚᣺ືゎ
ゎᯒࣔࢹࣝࢆ
ࢆ㧗⢭ᗘ໬ࡍ
ࡍࡿ᪉ἲ࡛࠶
࠶ࡿ㸬ṧࡿ୍
୍ࡘࡣࡡࡌࡾ
ࡾ᣺ືゎᯒ࡟
࡟㛵ࡍ
ࡿࡶࡢ
ࡢ࡛࠶ࡾ㸪㍈┤ᚄኚ໬㒊
㍈
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧࢆ⪃៖
៖ࡋࡓࣔࢹࣝ໬
໬᪉ἲ࡛࠶ࡿ
ࡿ㸦ᅗ 1.6 ཧ↷㸧㸬
࡞࠾
࠾㸪1 ḟඖࡢ
ࡢ㍈せ⣲࡛ࣔ
ࣔࢹࣝ໬ࡉࢀ ࡿ෇ᙧ᩿㠃ࡢ୰ᐇ㍈ࢆᣢ
ᣢࡘ➼᪉ᛶࣟ
࣮ࣟࢱࢆᑐ㇟
㇟࡜ࡋ㸪
ᅇ㌿㍈
㍈࡟ࡣᅇ㌿㍈
㍈᪉ྥࡢᘬᙇ
ᙇ࣭ᅽ⦰ຊ࡜
࡜ᅇ㌿᪉ྥࡢ
ࡢࢺࣝࢡࡀస
స⏝ࡋ࡚࠸࡞
࡞࠸ࡶࡢ࡜ࡍ
ࡍࡿ㸬
ࡉࡽ࡟
࡟㸪ᙎᛶ⠊ᅖ
ᅖ࡟ไ㝈ࡀ࡞
࡞࠸ࡶࡢ࡜ࡍ
ࡍࡿ㸬
ࡲࡓ
ࡓ㸪ᮏ◊✲࡛
࡛ࡣ㸪᭤ࡆ᣺
᣺ື࡜ࡡࡌࡾ
ࡾ᣺ືࢆศࡅ
ࡅ࡚㆟ㄽࡍࡿ
ࡿࡓࡵ㸪᭤ࡆ
ࡆ࡜ࡡࡌࡾࡢ
ࡢ㐃ᡂ
᣺ືࡣ
ࡣ⪃៖ࡢᑐ㇟
㇟እ࡜ࡍࡿ㸬
8
Accuracy improvement
of 1D rotordynamic analysis model
Considering the effect of the elasticity
in the face of the stepped cross-sections
Chapter 3
Bending vibration analysis
Considering shrink fit effect
for the bending stiffness
Chapter 4
Identifying all characteristic matrices
in the vibration equation of a rotor
based on the rigid and elastic properties
which can be obtained by a vibration test
Chapter 6
Considering the effect of the elasticity
in the face of the stepped cross-sections
Torsional vibration analysis
Chapter 5
Fig. 1.6
Overview of the study
➨ 2 ❶࡛ࡣ㸪1 ḟඖᱱࡢ᭷㝈せ⣲ἲࡀ㐺⏝ࡉࢀࡓ࣮ࣟࢱࡢ᭤ࡆ᣺ືゎᯒ࡜ࡡࡌࡾ᣺ືゎᯒ
࡟࠾ࡅࡿࣔࢹࣝ໬ࡢ᪉ἲ࡟ࡘ࠸࡚㏙࡭ࡿ㸬
➨ 3 ❶࡛ࡣ㸪㍈┤ᚄኚ໬㒊ࡢ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ໬᪉ἲࢆᥦ᱌ࡍࡿ㸬ࡲࡎ㸪኱ᚄ㒊࡜ᑠᚄ
㒊ࡢ┤ᚄẚ࠾ࡼࡧ኱ᚄ㒊ࡢ┤ᚄ࡜㍈᪉ྥ㛗ࡉ࡟㛵ࡍࡿᥦ᱌᪉ἲࡢ㐺⏝⠊ᅖ࡟ࡘ࠸᳨࡚ウࡍ
ࡿ㸬ḟ࡟㸪ᥦ᱌᪉ἲࡢヲ⣽ࢆ♧ࡍ㸬ࡑࡋ࡚㸪ᥦ᱌᪉ἲࡢጇᙜᛶࢆ᳨ドࡍࡿࡓࡵ࡟㸪ᥦ᱌᪉
ἲࢆ㐺⏝ࡋࡓ 1 ḟඖᱱࣔࢹࣝࡢゎᯒ⤖ᯝࢆ㸪3 ḟඖࣔࢹࣝࡢゎᯒ⤖ᯝ࠾ࡼࡧᐇ㦂⤖ᯝ࡜ẚ㍑
ࡋ㸪᭱ᚋ࡟㸪ᐇᶵ┦ᙜࡢᅇ㌿ᶵᲔ࡜ࡋ࡚⵨Ẽࢱ࣮ࣅࣥⓎ㟁ᶵࢆྲྀࡾୖࡆ࡚㸪ᥦ᱌᪉ἲࢆ㐺
⏝ࡋࡓሙྜ࡜㐺⏝ࡋ࡞࠸ሙྜࡢᅛ᭷್ࡢᕪ␗ࢆ♧ࡋ࡚㸪ᥦ᱌᪉ἲ࡟ࡼࡿ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ
໬ࡢຠᯝࢆ☜ㄆࡍࡿ㸬
➨ 4 ❶࡛ࡣ㸪ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࢆ࡝ࡢ⛬ᗘ࡟ぢ✚ࡶࡿࡢࡀጇᙜ࡛࠶ࡿ࠿࡟ࡘ࠸᳨࡚
ウࡋࡓ⤖ᯝࢆ㏙࡭ࡿ㸬ࡲࡎ㸪ᔐࡵྜ࠸㒊ࡢᦶ᧿ࢆ⪃៖ࡋࡓ᭷㝈せ⣲ゎᯒ࡟ࡼࡗ࡚ᔐࡵྜ࠸
㒊ࡢ᭤ࡆ๛ᛶࢆ᳨ウࡍࡿ㸬ḟ࡟㸪ᐇ㦂࡟ࡼࡿ᳨ド⤖ᯝࢆ♧ࡍ㸬
➨ 5 ❶࡛ࡣ㸪㍈┤ᚄኚ໬㒊ࡢࡡࡌࡾ๛ᛶࡢ㧗⢭ᗘ໬᪉ἲࢆᥦ᱌ࡍࡿ㸬ࡲࡎ㸪ᥦ᱌᪉ἲࡢ
㐺⏝⠊ᅖ࡟ࡘ࠸᳨࡚ウࡍࡿ㸬ḟ࡟㸪ᥦ᱌᪉ἲࡢヲ⣽ࢆ♧ࡍ㸬ࡑࡋ࡚㸪ᥦ᱌᪉ἲࡢጇᙜᛶࢆ
᳨ドࡍࡿࡓࡵ࡟㸪ᥦ᱌᪉ἲࢆ㐺⏝ࡋࡓ㸯ḟඖᱱࣔࢹࣝࡢゎᯒ⤖ᯝࢆ 3 ḟඖࣔࢹࣝࡢゎᯒ⤖
ᯝ࡜ẚ㍑ࡍࡿ㸬᭱ᚋ࡟㸪ᐇᶵ┦ᙜࡢᅇ㌿ᶵᲔ࡜ࡋ࡚⵨Ẽࢱ࣮ࣅࣥⓎ㟁ᶵࢆྲྀࡾୖࡆ࡚㸪ᥦ
᱌᪉ἲࢆ㐺⏝ࡋࡓሙྜ࡜㐺⏝ࡋ࡞࠸ሙྜࡢᅛ᭷್ࡢᕪ␗ࢆ♧ࡋ㸪ᥦ᱌᪉ἲ࡟ࡼࡿࡡࡌࡾ๛
9
ᛶࡢ㧗⢭ᗘ໬ࡢຠᯝࢆ☜ㄆࡍࡿ㸬
➨ 6 ❶࡛ࡣ㸪᣺ືヨ㦂⤖ᯝ࡟ᇶ࡙࠸ࡓ㍈᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬᪉ἲ࡟ࡘ࠸࡚㏙࡭ࡿ㸬
ࡲࡎ㸪≉ᛶ⾜ิࡢྠᐃ᪉ἲࡢᴫせࢆ㏙࡭㸪ḟ࡟ලయⓗ࡞ྠᐃᡭ㡰ࢆㄝ᫂ࡍࡿ㸬ࡑࡢḟ࡟㸪
๛య≉ᛶ࡜ᙎᛶ≉ᛶ࡟ㄗᕪࢆᣢࡘゎᯒࣔࢹࣝࡢලయⓗ࡞㧗⢭ᗘ໬౛࡜ࡋ࡚㸪ᶍᨃࡢᐇ ್
࡟ࡼࡗ࡚ 4 ẁࡢ㐲ᚰᅽ⦰ᶵ࣮ࣟࢱࡢࣔࢹࣝࡢ≉ᛶ⾜ิࢆྠᐃࡋࡓ⤖ᯝࢆ♧ࡍ㸬
୍ࡘ┠ࡢㄢ㢟࡛࠶ࡿ㸪ィ⟬㈇Ⲵࡀᑠࡉࡃ㧗⢭ᗘ࡞㍈᣺ື㸦᭤ࡆ᣺ື࡜ࡡࡌࡾ᣺ື㸧ゎᯒ
ᡭἲࡢ㛤Ⓨ࡟࠾࠸࡚㸪᭤ࡆ᣺ືゎᯒ࡟㛵ࡋ࡚ࡣ㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋࡓࣔ
ࢹࣝ໬᪉ἲ࡜ᔐࡵྜ࠸㒊ࡢࣔࢹࣝ໬᪉ἲࢆ☜❧ࡋ㸪ࡡࡌࡾ᣺ືゎᯒ㛵ࡋ࡚ࡣ㍈┤ᚄኚ໬㒊
᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋࡓࣔࢹࣝ໬᪉ἲࢆ☜❧ࡋࡓ㸬᭤ࡆ᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬࡛ࡣ㸪
㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋࡓࣔࢹࣝ໬᪉ἲ࡜ࡋ࡚㸪㍈┤ᚄኚ໬㒊ࡢᑠᚄ㍈ഃ࡟
㐺ษ࡞㛗ࡉࡢせ⣲ࢆタࡅ࡚㸪ࡑࡢせ⣲ࡢ᭤ࡆ๛ᛶࢆ㐺ษ࡟ᙅࡵࡿ᪉ἲࢆ☜❧ࡋ㸪ᔐࡵྜ࠸
㒊ࡢࣔࢹࣝ໬᪉ἲ࡜ࡋ࡚ࡣ㸪ᔐࡵྜ࠸㒊ࡢ➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿ㍈ࡢእᚄࢆ㸪ᔐࡵྜࢃ
ࡏࡿࢫ࣮ࣜࣈࡢእᚄ࡜ࡍࡿ᪉ἲࢆ♧ࡋࡓ㸬ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬࡛ࡣ㸪㍈┤ᚄ
ኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋࡓࣔࢹࣝ໬᪉ἲ࡜ࡋ࡚㸪㍈┤ᚄኚ໬㒊ࡢᑠᚄ㍈ഃ࡟㐺ษ࡞
㛗ࡉࡢせ⣲ࢆタࡅ࡚㸪ࡑࡢせ⣲ࡢࡡࡌࡾ๛ᛶࢆ㐺ษ࡟ᙅࡵࡿ᪉ἲࢆ☜❧ࡋࡓ㸬
஧ࡘ┠ࡢㄢ㢟࡛࠶ࡿ㸪ᐇ㦂࣮ࣔࢻゎᯒ⤖ᯝ࡟ᇶ࡙࠸࡚࣮ࣟࢱࡢ᣺ືゎᯒࣔࢹࣝࢆ㧗⢭ᗘ
໬ࡍࡿ᪉ἲࡢ㛤Ⓨ࡟ᑐࡋ࡚ࡣ㸪᭤ࡆ᣺ືゎᯒࣔࢹࣝ࡟ࡘ࠸࡚㸪㉁㔞㸪㔜ᚰ఩⨨㸪័ᛶ࣮ࣔ
࣓ࣥࢺ࡜㸪ᅛ᭷᣺ືᩘ㸪ᅛ᭷࣮ࣔࢻ࡟㛵ࡍࡿ᝟ሗ࠿ࡽ㸪ㄗᕪࢆྵࢇࡔ඲࡚ࡢᮦᩱ≉ᛶ್ࢆ
ྠ᫬࡟㧗⢭ᗘ໬ࡋ࡚㸪᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ඲࡚ࡢ≉ᛶ⾜ิ㸦㉁㔞⾜ิ㸪ῶ⾶⾜ิ㸪๛ᛶ⾜
ิ㸪࠾ࡼࡧࢪࣕ࢖ࣟ⾜ิ㸧ࢆྠᐃࡍࡿ᪉ἲࢆ☜❧ࡋࡓ㸬
ཧ⪃ᩥ⊩
Balmes, E., Frequency domain identification of structural dynamics using the pole/residue
parametrization, International Modal Analysis Conference (1996).
Ewins, D. J., Modal testing, theory, practice, and application, John Wiley and Sons, Inc. (1984).
ᯘ ᕑ, ᒾ㝃 ಙ⾜, ∾ Ὀᕼ, ㍈୍య෇ᯈࡢᅛ᭷᣺ື࣮ࣔࢻ࡟ࡘ࠸࡚㸦➨ 2 ሗ ෇ᯈ࡜㍈ࡢቃ
⏺㒊࡟࠾ࡅࡿᙎᛶቃ⏺᮲௳ᘧࡢࡤࡡᐃᩘ㸧, ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟 C ⦅, Vol.56, No.524
(1990), pp. 952-957.
Ishida, Y. and Yamamoto, T., Liner and nonliner rotordynamics – A modern treatment with
applications (2nd enlarged and improved edition), WILEY-VCH (2012), pp. 351-371.
▼ཎ ᅜᙪ, ⏣୰ 㐨㞝, ᒣᓮ ὒᩥ, ࢱ࣮ࣅࣥື⩼࡟ຍࢃࡿὶయບ᣺ຊ࡟㛵ࡍࡿ◊✲ : ➨ 1
ሗ, ᐇ㦂᪉ἲ࡜⤖ᯝ, ᪥ᮏᶵᲔᏛ᭳ㄽᩥ㞟 B ⦅, Vol. 56, No. 528 (1990), pp. 2390-2396.
ᒾና ༟୕, ࣮ࣟࢱ࣭㍈ཷ⣔ࡢ᣺ື࡟ཬࡰࡍㄗᕪࡢᙳ㡪, ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟 C ⦅, Vol. 46,
No.405, (1980), pp. 493-500.
ᒾና ༟୕, ᮏᒸ ┤ே, ᕝ஭ Ⰻḟ, ࣛࣅࣜࣥࢫࢩ࣮ࣝࢆ᭷ࡍࡿᅇ㌿㍈ࡢ୙Ᏻᐃ᣺ື : ➨ 1
ሗ,ᾮయࣔࢹࣝ࡟ࡼࡿࢩ࣮ࣝෆὶయຊࡢᐇ㦂࠾ࡼࡧ⌮ㄽ࡟ࡼࡿゎᯒ, ᪥ᮏᶵᲔᏛ఍ㄽᩥ
㞟 C ⦅, Vol. 49, No.441, (1983), pp. 719-726.
ຍᕝ ᖾ㞝, ▼ᕝ ṇ⮧, ࣮ࣔࢲࣝゎᯒධ㛛, ࣮࣒࢜♫ (1987).
㔠Ꮚ ᗣᬛ, 㛫℩ ṇ㝯, ⸨⏣ ຾ஂ, 㛗ᔱ ฼ኵ, ࣑ࢫࢳ࣮ࣗࣥࡀ࠶ࡿ⩼࣭ࢹ࢕ࢫࢡ⣔ࡢᛂ⟅
ゎᯒ, ᪥ᮏᶵᲔᏛ᭳ㄽᩥ㞟 C ⦅, Vol. 58, No. 547 (1992), pp. 744-749.
10
⚄ྜྷ ༤, ᕝす ㏻⿱, ᑠ㔝 ⪔ྖ, LMI ᭱㐺໬ࢆ⏝࠸ࡓ࣮ࣟࢱࣂࣛࣥࢩࣥࢢἲ, ᪥ᮏᶵᲔᏛ᭳
ㄽᩥ㞟 C ⦅, Vol. 65, No. 634 (1999), pp. 2218-2225.
⳥ᆅ ຾᫛, ከ㍈ཷከ෇ᯈᅇ㌿㍈⣔ࡢ୙ࡘࡾ࠶࠸᣺ືࡢゎᯒ, ᪥ᮏᶵᲔᏛ᭳ㄅ, Vol. 72,
No.610 (1969), pp. 1556-1563.
ᑠᇼ ୚୍, ࢱ࣮ࣅࣥ㌴㍈ࡢ᭤ࡆࡇࢃࡉ࡜ᅛ᭷᣺ືᩘࡢᶍᆺᐇ㦂, ᪥ᮏᶵᲔᏛ఍ㄅ, Vol.32,
No.243 (1966), pp.1675-1681.
Lalanne, M. and Ferraris, G., Rotordynamics prediction in engineering (second edition), pp.
132-134, JOHN WILEY & SONS (1998).
Maia, N. M. M., Silva, J. M. M., Theoretical and experimental modal analysis, John Wiley & Sons,
Inc. (1997).
ᯇୗ ಟᕬ, ஭⏣ 㐨⛅, ᙎᛶ࣮ࣟࢱࡢᅇ㌿ḟᩘẚᛂ⟅ゎᯒ, ᪥ᮏᶵᲔᏛ఍ㄽᩥ C ⦅, Vol. 50,
No. 452 (1984), pp. 626-634.
ᯇୗ ಟᕫ, ᑠᯘ ṇ⏕, ྂụ ἞Ꮥ, ⚄ྜྷ ༤, ⏣୰ ṇே, ⥆ ᅇ㌿ᶵᲔࡢ᣺ືʊᐇᶵࡢ᣺ື
ၥ㢟࡜᣺ືデ᩿, ࢥࣟࢼ♫ (2012), p. 287
ᯇୗ ಟᕭ, 㛵ཱྀ ᾈ୍, 㡲⸨ ṇᗤ, ⱂἑ ᖾ⏨, ࣐ࣝࢳࣂࣛࣥࢫᅽ⦰ᶵࡢ㟼⢔໬, ᪥ᮏᶵᲔ
Ꮫ᭳ㄽᩥ㞟 C ⦅, Vol. 58, No. 550 (1992), pp. 1848-1853.
ᯇୗ ಟᕫ, ⏣୰ ṇே, ⚄ྜྷ ༤, ᑠᯘ ṇ⏕, ᅇ㌿ᶵᲔࡢ᣺ືʊᐇ⏝ⓗ᣺ືゎᯒࡢᇶᮏ, ࢥ
ࣟࢼ♫ (2009), p. 97.
୕㍯ ಟ୕, ᙎᛶ࣮ࣟࢱࡢࡘࡾ࠶ࢃࡏ, ᪥ᮏᶵᲔᏛ᭳ㄽᩥ㞟, Vol.37, No.297 (1971), pp.
949-958.
㛗ᯇ ᫛⏨, ࣮ࣔࢻゎᯒධ㛛, ࢥࣟࢼ♫ (1993).
すᔱ つୡ, 㐲⸨ ᙲ, ᒣཱྀ ࿴ᖾ, ᅇ㌿ᶵᲔࡢ㍈ࢩ࣮࡛ࣝⓎ⏕ࡍࡿ୙Ᏻᐃὶయຊࡢᩘ್ゎᯒ
㸦ࢩ࣮ࣝ㛗ࡉ࡜ࢠࣕࢵࣉࡢᙳ㡪㸧, ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟, Vol.80 (2014), No.816.
Pestel, C. and Leckie, F. A., Matrix Method in Elasto Mechanics, McGraw-Hill (1963)
ሷᖭ ᏹつ, ⸨ἑ ஧୕ኵ, ከ㍈ཷᅇ㌿㍈⣔ࡢࡘࡾ࠶ࢃࡏἲ : ➨ 1 ሗ, ከ㠃ከ㏿ᗘࡘࡾ࠶ࢃ
ࡏ, ᪥ᮏᶵᲔᏛ᭳ㄽᩥ㞟 C ⦅, Vol. 45, No.391 (1979), pp. 275-284.
ⓑᮌ ୓༤, ⚄ྜྷ ༤, Ᏻ⏣༓⛅, ᪂ࢥࣥࣆ࣮ࣗࢱࣂࣛࣥࢫࢩࢫࢸ࣒ࡢ㛤Ⓨ, ୕⳻㔜ᕤᢏሗ,
17 ᕳ, 2 ྕ (1980), pp. 120-129.
㧗⏣ ᫀ஧, ⯚➉ Ⰻ㞝, ✄ᇉ ჆அ, ㍈┤ᚄኚ໬㒊ࢆ᭷ࡍࡿ࣮ࣟࢱࡢ᭤ࡆ᣺ື࣮ࣔࢻ࡟㛵ࡍ
ࡿ᣺ືィ⟬ࡢ㧗⢭ᗘ⡆᫆ࣔࢹࣝࡢᥦ᱌, ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟 C ⦅, Vol.75, No.752 (2009),
pp.824-829.
ෆᾏ ᨻ᫓, ᕝᓮ ⪽, ℧⏣ ⣧ஓ, ᓥᇉ ‶, Ᏻ㐩 ࿴ᙪ, ஭ୖ ๛ᚿ, ࣔࣥࢸ࢝ࣝࣟἲ࡜ᐇ㦂
ィ⏬ἲ࡟ࡼࡿࣟࢣࢵࢺࢱ࣮࣏࣎ࣥࣉ⏝࣮ࣟࢱࡢ㍈ᙧ≧ࡢ᭱㐺タィ, ᪥ᮏᶵᲔᏛ఍
Dynamics and Design Conference 2013 ㅮ₇ㄽᩥ㞟 (2013).
▮㘠 㔜ኵ, ⏣ᮧ ❶⩏, 2 ෇ᯈᙎᛶ࣮ࣟࢱࡢከ㠃ࡘࡾ࠶ࢃࡏ, ᪥ᮏᶵᲔᏛ᭳ㄽᩥ㞟, Vol.35,
No.272 (1969), pp. 742-750.
ྜྷᮏ ሀ୍, ᯇୗ ಟᕫ, Mathematica ࡛Ꮫࡪ᣺ື࡜ࢲ࢖ࢼ࣑ࢡࢫࡢ⌮ㄽ, ᳃໭ฟ∧ (2004), p.
31.
Zutavern, Z. S. and Childs, D. W., Identification of rotordynamic forces in a flexible fotor system
using magnetic bearings, Transactions of the ASME Vol. 130 (2008), pp. 022504/1-6.
11
➨❶
❶ ࣮ࣟࢱࡢ
ࡢ᭤ࡆ᣺ືゎ
ゎᯒ࡜ࡡࡌࡾ
ࡾ᣺ືゎᯒࡢ
ࡢࡓࡵࡢ ḟඖᱱࡢ᭷㝈
ḟ
㝈せ⣲ࣔࢹࣝ
ࣝ
⥴ゝ
◊✲࡛ࡣ㸪୰
୰ᐇ㍈ࢆᑐ㇟
㇟࡜ࡋࡓ㸪1 ḟඖᱱࡢ᭷㝈せ⣲ἲࢆ㐺
㐺⏝ࡋࡓ᭤ࡆ
ࡆ᣺ືゎᯒࣔ
ࣔࢹࣝ
ᮏ◊
࡜ࡡࡌࡾ᣺ືゎᯒ
ᯒࣔࢹࣝࡢ㧗
㧗⢭ᗘ໬ࢆ┠
┠ⓗ࡜ࡋ࡚࠸
࠸ࡿ㸬ࡑࡇ࡛
࡛㸪ᮏ❶࡛ࡣ
ࡣ࣮ࣟࢱࡢ᭤
᭤ࡆ᣺
ࡡࡌࡾ᣺ືࡢ
ࡢ 1 ḟඖᱱࡢ
ࡢ᭷㝈せ⣲ゎ
ゎᯒࣔࢹࣝࡢ
ࡢࣔࢹࣝ໬᪉
᪉ἲ࡟ࡘ࠸࡚
࡚ᴫㄝࡍࡿ㸬
ື࡜ࡡ
2.22 ⠇࡛ࡣ㸪᭤
᭤ࡆ᣺ືࡢゎ
ゎᯒࣔࢹࣝ࡟ ࡘ࠸࡚㸪ᑐ㇟
㇟࡜ࡍࡿ⮬⏤
⏤ᗘ㸪㐠ື᪉
᪉⛬ᘧ㸪ᅇ㌿
㌿㍈ࡢ
᭷㝈せ
せ⣲ࣔࢹࣝ㸪
㸪ࢹ࢕ࢫࢡࡢ
ࡢ᭷㝈せ⣲ࣔ
ࣔࢹࣝ㸪࣮ࣟ
࣮ࢱ඲యࡢ᭷
᭷㝈せ⣲ࣔࢹ
ࢹࣝ࡟ࡘ࠸࡚
࡚ㄝ᫂
ࡍࡿ㸦㛗Ụ, ᚋ⸨
⸨, 2013㸧㸬
ࡡࡌࡾ᣺ືࡢ
ࡢゎᯒࣔࢹࣝ ࡟ࡘ࠸࡚㸪ᑐ
ᑐ㇟࡜ࡍࡿ⮬
⮬⏤ᗘ㸪㐠ື
ື᪉⛬ᘧ㸪ᅇ
ᅇ㌿㍈
2.33 ⠇࡛ࡣ㸪ࡡ
ࡢ᭷㝈
㝈せ⣲ࣔࢹࣝ
ࣝ㸪ࢹ࢕ࢫࢡ
ࢡࡢ᭷㝈せ⣲
⣲ࣔࢹࣝ㸪ࣟ
࣮ࣟࢱ඲యࡢ
ࡢ᭷㝈せ⣲ࣔ
ࣔࢹࣝ࡟ࡘ࠸
࠸࡚ㄝ
᫂ࡍࡿ㸬
᭤ࡆ᣺ືࡢ
ࡢゎᯒࣔࢹࣝ
ࣝ
ືゎᯒࣔࢹࣝ
ࣝࡢ⮬⏤ᗘ
᭤ࡆ᣺ື
㍈࡜ࢹ࢕ࢫࢡ࡛
࡛ᵓᡂࡉࢀࡿ
ࡿ࣮ࣟࢱࡣ㸪 n ಶࡢ⠇Ⅼ
Ⅼࢆᣢࡕ㸪ྛ⠇Ⅼࡣᅗ 2.11 ࡢᗙᶆ⣔࡟
࡟࠾࠸
୪㐍ኚ఩࡜ x㸪y ㍈࿘ࡾࡢ
ࡢᅇ㌿ゅ șx㸪șy ࡢィ㸲ࡘ
ࡘࡢ⮬⏤ᗘࢆ
ࢆᣢࡘࡶࡢ࡜ࡍࡿ㸬
࡚ x㸪y ᪉ྥࡢ୪
㸪ᅇ㌿ゅࡣྛ
ྛᗙᶆ㍈ࡢྑ
ྑࡡࡌ᪉ྥࢆ
ࢆṇ࡜ࡍࡿ㸬
࡞࠾㸪
Figg. 2.1
Exam
mple of a rotoor model and definition of coordinate syystem
᭤ࡆ᣺ື
ືゎᯒࣔࢹࣝ
ࣝࡢ㐠ື᪉⛬
⛬ᘧ
ᮏ◊
◊✲࡛ࡣ㸪᭤
᭤ࡆ᣺ືゎᯒ
ᯒ࡟⏝࠸ࡿࣔ
ࣔࢹࣝ࡜ࡋ࡚
࡚ Friswell, M.
M I.ࡽ㸦Frisw
well, et al., 2010㸧
ࡢࡶࡢ
ࡢࢆ⏝࠸ࡿ㸬㍈ཷࡸࢩ࣮
࣮ࣝ㒊ࡢ௜ຍ
ຍ㉁㔞ࡀ↓ど
ど࡛ࡁࡿሙྜ
ྜ㸪㍈ཷ࡛ᨭᣢ
ᣢࡉࢀ㸪ᅇ㌿
㌿ᩘ ȍ
࡛ᅇ㌿
㌿ࡍࡿ࣮ࣟࢱ
ࢱࡢ᭤ࡆ᣺ື
ືࡢ㐠ື᪉⛬
⛬ᘧࡣᘧ㸦2.1㸧࡛⾲ࡉࢀ
ࢀࡿ㸬ࡇࡇ࡛
࡛㸪[ ]ࡣ⾜ิ
ิࢆ⾲
ࡋ㸪{ }ࡣ࣋ࢡࢺࣝ
ࣝࢆ⾲ࡍ㸬
12
>M r @^X ` >Cr @ >Cb @ :>G r @^X ` >K r @ >Kb @^X ` ^F ` 㸦2.1㸧
[Mr]㸸࣮ࣟࢱࡢ㉁㔞⾜ิ㸪ࢧ࢖ࢬ 4n×4n㸪ᑐ⛠⾜ิ
[Cr]㸸࣮ࣟࢱࡢῶ⾶⾜ิ㸪ࢧ࢖ࢬ 4n×4n㸪ᑐ⛠⾜ิ
[Cb]㸸㍈ཷࡢῶ⾶⾜ิ㸪ࢧ࢖ࢬ 4n×4n
[Gr]㸸࣮ࣟࢱࡢࢪࣕ࢖ࣟ⾜ิ㸪ࢧ࢖ࢬ 4n×4n㸪ṍᑐ⛠⾜ิ
[Kr]㸸࣮ࣟࢱࡢ๛ᛶ⾜ิ㸪ࢧ࢖ࢬ 4n×4n㸪ᑐ⛠⾜ิ
[Kb]㸸㍈ཷࡢ๛ᛶ⾜ิ㸪ࢧ࢖ࢬ 4n×4n
{X}㸸ኚ఩࣋ࢡࢺࣝ㸪ࢧ࢖ࢬ 4n×1㸪
^X ` ^x1 , y1 ,T x1 ,T y1 , x2 , y2 ,T x 2 ,T y 2 ,..., xn , yn ,T xn ,T yn `T 㸪T ࡣ㌿⨨ࢆ⾲ࡍグྕ
{F}㸸ຊ࣋ࢡࢺࣝ㸪ࢧ࢖ࢬ 4n×1㸪
^F `
^f
`
T
x1
, f y1 , M T x 1 , M T y 1 , f x 2 , f y 2 , M T x 2 , M T y 2 ,..., f xn , f yn , M T x n , M T y n 㸪
f ࡣຊ (N)㸪M ࡣ࣮࣓ࣔࣥࢺ (Nm)
㍈ཷࡢ๛ᛶ⾜ิ[Kb]࡜ῶ⾶⾜ิ[Cb]ࡣ㸪ࢧ࢖ࢬ 4n×4n ࡢࢮࣟ⾜ิࡢ㍈ཷࡢᙜヱ⠇Ⅼ㒊ศࡢ
せ⣲࡟ࡑࢀࡒࢀ㍈ཷࡢ๛ᛶ࡜ῶ⾶ࡀᐃ⩏ࡉࢀࡓࡶࡢ࡛࠶ࡿ㸬ᘧ㸦2.1㸧࡛⾲ࡉࢀࡿ࣮ࣟࢱ㸫
㍈ཷ⣔ࡢ༴㝤㏿ᗘࡸᏳᐃᛶ࡞࡝ࢆゎᯒ࡟ࡼࡗ࡚ホ౯ࡍࡿሙྜ㸪ゎᯒ⤖ᯝࡢ⢭ᗘࡣ㸪࣮ࣟࢱ
ࡢ≉ᛶ⾜ิ[Mr]㸪[Cr]㸪[Kr]㸪[Gr]࡜㍈ཷ≉ᛶ[Kb]㸪[Cb]ࡢ⢭ᗘ࡟౫Ꮡࡍࡿ㸬
⮬⏤ᨭᣢࡉࢀࡓ㠀ᅇ㌿᫬ࡢ࣮ࣟࢱࡢ㐠ື᪉⛬ᘧࡣ㸪ᘧ㸦2.1㸧 ࠿ࡽ㍈ཷ๛ᛶ㡯[Kb]㸪㍈ཷ
ῶ⾶㡯[Cb]㸪࣮ࣟࢱࡢࢪࣕ࢖ࣟ㡯[Gr]ࢆྲྀࡾ㝖࠸࡚ᘧ㸦2.2㸧࡛⾲ࡉࢀࡿ㸬
>M r @^X ` >Cr @^X ` >K r @^X ` ^F ` 㸦2.2㸧
᭤ࡆ᣺ືゎᯒࣔࢹࣝ࡟࠾ࡅࡿᅇ㌿㍈ࡢࣔࢹࣝ໬
ᮏ◊✲࡛ࡣ㸪ࡏࢇ᩿ኚᙧ࡜ᅇ㌿័ᛶ㸦ࡓࢃࡳ᭤⥺ࡢഴࡁ࡟క࠺័ᛶ࣮࣓ࣔࣥࢺ㸧ࢆ⪃៖
ࡋࡓࢸ࢕ࣔࢩ࢙ࣥࢥᱱ࡛ᅇ㌿㍈ࡢせ⣲ࢆࣔࢹࣝ໬ࡍࡿ㸬ࡲࡓ㸪୍⯡࡟௜㘓 A-1 ࡟♧ࡉࢀࡿ
ࡼ࠺࡟ᐦᗘ㸪࣏࢔ࢯࣥẚ㸪ࣖࣥࢢ⋡࡞࡝ࡢᮦᩱ≉ᛶ್࡛ᐃ⩏ࡉࢀࡿ㐠ື᪉⛬ᘧ࡟࠾ࡅࡿ≉
ᛶ⾜ิࡢせ⣲ࢆ㸪ᐃ⩏ᘧࡢ↹㞧ࡉࢆపῶࡍࡿࡓࡵ࡟㸪᭤ࡆ๛ᛶ࡜᪂ࡓ࡟ᑟධࡋࡓ㉁㔞ಀᩘ
࡜័ᛶಀᩘ࡞࡝࡟ࡼࡗ࡚ᐃ⩏ࡍࡿ㸬
ᐦᗘ ȡ (kg/m3)㸪࣏࢔ࢯࣥẚ Ȟ㸪እᚄ de (m)㸪㛗ࡉ le (m)ࡢ㍈せ⣲ࡢ㉁㔞ಀᩘ Įm ࡜័ᛶಀᩘ
Įi ࢆࡑࢀࡒࢀᘧ㸦2.3㸧࡜ᘧ㸦2.4㸧࡛ᐃ⩏ࡍࡿ㸬
13
Dm
UAe le
USd e 2le
840
3360
㸦2.3㸧
Ae㸸᩿㠃✚ (m2)㸪Ae=ʌde2/4
Di
UI e
USd e 4
30le
1920le
㸦2.4㸧
Ie㸸᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ (m4)㸪Ie=ʌde4/64
ࡏࢇ᩿ಀᩘ IJe ࡣᘧ㸦2.5㸧࡛ᐃ⩏ࡉࢀࡿ㸬
We
12 EI e
7 6Q d e 2
NGAe le 2
4le
㸦2.5㸧
2
E㸸ࣖࣥࢢ⋡ (N/m2)
ț㸸ࡏࢇ᩿ᐃᩘ㸦↓ḟඖ㸧㸪ț=6(1+Ȟ)/(7+6Ȟ)
Ȟ㸸࣏࢔ࢯࣥẚ㸦↓ḟඖ㸧
G㸸ᶓᙎᛶಀᩘ (N/m2)㸪G= E/{2(1+Ȟ)}
ᘧ㸦2.6㸧࡟♧ࡉࢀࡿࡼ࠺࡟㸪㍈せ⣲ࡢ㉁㔞⾜ิ[Mse]ࡣࢧ࢖ࢬ 8×8 ࡢᑐ⛠⾜ิ࡛࠶ࡾ㸪[Mse]
ࡢせ⣲ mij㸦i=1,2,...,8, j=1,2,...,8㸧ࡣ㉁㔞ಀᩘ Įm㸪័ᛶಀᩘ Įi㸪ࡏࢇ᩿ಀᩘ IJe㸪࠾ࡼࡧ㛗ࡉ le
ࡢ㛵ᩘ࡜ࡋ࡚ᐃ⩏ࡉࢀࡿ㸬
>M se @
m12
m18 º
m 28 »»
㸦2.6㸧
»
»
m88 ¼
ª m11 m12
«
m 22
«
«
«
¬ Sym.
m13
m16
m17
m68
m78
0
m11
m22
m55
m66
m14
m23
m15
m26
m18
m27
^
m58
^D
m
m36
m24
^D
m67
m
m25
m28
m34
m35
m38
m46
m47
^D
m
1 W e 2 `˜ 44 77W 35W e 2 ˜ le ^D i 1 W e 2 `˜ 3 15W e ˜ le
m45
`
m57
1 W e 2 `˜ 312 558W e 280W e 2 ^D i 1 W e 2 `˜ 36
1 W e 2 `˜ 108 252W e 140W e 2 ^D i 1 W e 2 `˜ 36 ^
`
D m 1 W e ˜ 26 63W e 35W e ˜ le D i 1 W e ˜ 3 15W e ˜ le
2
m56
2
14
2
^D
m88
1 W e 2 `˜ 8 14W e 7W e 2 ˜ le 2 ^D i 1 W e 2 `˜ 4 5W e 10W e 2 ˜ le 2
m33
m44
m77
m37
m48
D m 1 W e ˜ 6 14W e 7W e ˜ le D i 1 W e ˜ 1 5W e 5W e ˜ le
^
2
m
`
2
^
2
2
`
2
2
ᘧ㸦2.7㸧࡟♧ࡉࢀࡿࡼ࠺࡟㸪㍈せ⣲ࡢ๛ᛶ⾜ิ[Kse]ࡣࢧ࢖ࢬ 8×8 ࡢᑐ⛠⾜ิ࡛࠶ࡾ㸪[Kse]
ࡢせ⣲ kij ࡣࡏࢇ᩿ಀᩘ IJe㸪᭤ࡆ๛ᛶ EIe㸦ࣖࣥࢢ⋡ E ࡜᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ Ie ࡢ✚㸧㸪࠾ࡼ
ࡧ㛗ࡉ le ࡢ㛵ᩘ࡜ࡋ࡚ᐃ⩏ࡉࢀࡿ㸬
>K se @
ª k11 k12 k18 º
«
k22 k28 »»
«
㸦2.7㸧
«
»
»
«
k88 ¼
¬ Sym.
k12
k13
k16
k17
k11
k 22
k 55
k14
k18
k 23
k 27
k 33
k 44
k 77
k 88
k 37
k 48
k 66
k 24
k 25
k15
k 28
k 34
k 26 12 EI e
k 45
k36
k 35
^1 W
k58
e
k 67
k 38
k 46
k 47
k 56
k 57
k 68
k 78
0
˜ le 3 `
6 EI e
^1 W ˜ l `
2
e
e
EI e 4 W e ^1 W e ˜ l e `
EI e 2 W e ^1 W e ˜ l e `
ᘧ㸦2.8㸧࡟♧ࡉࢀࡿࡼ࠺࡟㸪㍈せ⣲ࡢࢪࣕ࢖ࣟ⾜ิ[Gse]ࡣࢧ࢖ࢬ 8×8 ࡢṍᑐ⛠⾜ิ࡛࠶
ࡾ㸪[Gse]ࡢせ⣲ gij ࡣ័ᛶಀᩘ Įi㸪ࡏࢇ᩿ಀᩘ IJe㸪࠾ࡼࡧ㛗ࡉ le ࡢ㛵ᩘ࡜ࡋ࡚ᐃ⩏ࡉࢀࡿ㸬
>G se @
g11
g 12 g 18 º
ª g 11
«
g 22 g 28 »»
«
㸦2.8㸧
«
»
»
«
g 88 ¼
¬ Skew sym.
g14
g15
g18
g 22
g 23
g 26
g 58
g 66
g 67
g 77
g 88
0
g 46
g 57
g12
g 16
g13
g17
g 24
g 34
g 78
^2D
g 38
g 47
g 25
g 56
g 28
g 35
i
g 33
i
g 36
g 37
g 44
g 45
g 48
1 W 2 `˜ 36
g 68
1 W e 2 `˜ 4 5W e 10W e 2 ˜ le 2
i
^2D
^2D
g 27
1 W e 2 `˜ 1 5W e 5W e 2 ˜ le 2
15
^
`
2D i 1 W e ˜ 3 15W ˜ le
2
g 55
せ⣲ࡢῶ⾶⾜
⾜ิ[Cse]ࡣẚ
ẚ౛ῶ⾶ᐃᩘ
ᩘ Ȗ ࢆ⏝࠸࡚
࡚ᘧ㸦2.9㸧࡛
࡛㏆ఝࡉࢀࡿ
ࡿࡶࡢ࡜ࡍࡿ㸬
㍈せ
>C se @ >C r @
J >K se @ 㸦2.9㸧
㌿㍈඲యࡢ㉁
㉁㔞⾜ิ[Mshaft
]㸦ࢧ࢖ࢬ
ࢬ 4n×4n㸧ࡣ
ࡣ㸪ᅗ 2.2 ࡟♧
♧ࡍࡼ࠺࡟㸪
㸪1 ␒┠ࡢ㉁
㉁㔞⾜
ᅇ㌿
s
ิ[Mse1]࠿ࡽ n-1 ␒┠ࡢ㉁㔞⾜
␒
⾜ิ[Mse(n-1) ]ࢆ㔜ࡡྜࢃ
ࢃࡏࡿࡇ࡜࡟
࡟ࡼࡗ࡚ᵓᡂ ࡉࢀ㸪ᅇ㌿㍈
㍈඲య
ࡢ๛ᛶ
ᛶ⾜ิ[Kshaftt]㸪ῶ⾶⾜ิ
ิ[Cshaft]㸪࠾ࡼ
ࡼࡧࢪࣕ࢖ࣟ
ࣟ⾜ิ[Gshaft]ࡶྠᵝ࡟ࡋ࡚
࡚ᵓᡂࡉࢀࡿ
ࡿ㸬ࡇ
ࡇ࡛㸪
㸪㔜ࡡྜࢃࡏ
ࡏ࡜ࡣ༢⣧࿴
࿴࡛࠶ࡾ㸪ࡤ
ࡤࡡࡢ⤖ྜ᪉
᪉ἲ࡛ࡣ୪ิ
ิ⤖ྜ࡟┦ᙜ
ᙜࡍࡿ㸬
M se1
Mse2
Mse33
Mshaft=
Mse(n-1)
Fig. 2.2
Assembliing mass mattrix of beam element
e
᭤ࡆ᣺ື
ືゎᯒࣔࢹࣝ
ࣝ࡟࠾ࡅࡿࢹ
ࢹ࢕ࢫࢡࡢࣔࢹࣝ໬
ࢹ࢕ࢫࢡࡣ㍈ࡢ
ࡢ᭤ࡆ๛ᛶ࡟
࡟ᙳ㡪ࢆ୚࠼
࠼࡞࠸๛యせ
せ⣲࡛࠶ࡾ㸪㉁㔞࡜័ᛶ
ᛶ࣮࣓ࣔࣥࢺ
ࢺࡢࡳ
࣮ࢱࡢ᣺ື≉
≉ᛶ࡟ᙳ㡪ࢆ
ࢆཬࡰࡍࡶࡢ
ࡢ࡜ࡋ࡚ᢅ࠺
࠺㸬
ࡀ࣮ࣟ
1ࡘ
ࡘࡢࢹ࢕ࢫࢡ
ࢡࡢ᭷㝈せ⣲
⣲ࣔࢹࣝࡣࢧ
ࢧ࢖ࢬ 4×4 ࡢ㉁㔞⾜ิ
ࡢ
[Mdisc]࡜ࢧ࢖ࢬ
ࢬ 4×4 ࡢࢪࣕ
ࣕ࢖ࣟ
⾜ิ[Gdisc]ࡢࡳ࡛
࡛ᐃ⩏ࡉࢀࡿ㸬㉁㔞⾜ิ[ Mdisc]ࡢせ⣲
⣲ࡣᘧ㸦2.10㸧࡛ᐃ⩏ࡉࢀ
㸧
ࢀ㸪ࢪࣕ࢖ࣟ
ࣟ⾜ิ
[Gdisc]ࡣᘧ㸦2.11㸧࡛ᐃ⩏ࡉࢀ
㸧
ࢀࡿ㸬
>M disc @
ªmdiisc
« 0
«
« 0
«
¬ 0
0
mdisc
0
0
0
J disc
d
0
0
0 º
0 »»
0 »
»
J disc ¼
mdisc㸸ࢹ
ࢹ࢕ࢫࢡࡢ㉁
㉁㔞 (kg)
ࢹ࢕ࢫࢡࡢ┤
┤ᚄ㍈࿘ࡾࡢ័
័ᛶ࣮࣓ࣔࣥ
ࣥࢺ (kgm2)
Jdisc㸸ࢹ
16
㸦2.10㸧
>Gdisc @
ª0
«0
«
«0
«
¬«0
0
0
0
0
0
0
0 J pdisc
0 º
0 »»
㸦2.11㸧
J pdisc »
»
0 ¼»
Jpdisc㸸ࢹ࢕ࢫࢡࡢᅇ㌿㍈࿘ࡾࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ (kgm2)
ࢹ࢕ࢫࢡࡢ┤ᚄ㍈࿘ࡾࡢ័ᛶ࣮࣓ࣔࣥࢺ Jdisc (kgm2)࡜ࢹ࢕ࢫࢡࡢᅇ㌿㍈࿘ࡾࡢᴟ័ᛶࣔ
࣮࣓ࣥࢺ Jpdisc (kgm2)ࡣࡑࢀࡒࢀᘧ㸦2.12㸧࡜ᘧ㸦2.13㸧࡛⾲ࡉࢀࡿ㸬
J disc
D
64
S
4
out
Din4 U disc t disc 1
2
㸦2.12㸧
mdisc t disc
12
Dout㸸ࢹ࢕ࢫࢡࡢእᚄ (m)
Din㸸ࢹ࢕ࢫࢡࡢෆᚄ (m)
mdisc㸸ࢹ࢕ࢫࢡࡢ㉁㔞 (kg)㸪 mdisc
SU disc Dout 2 Din 2 ˜ t disc 4
ȡdisc㸸ࢹ࢕ࢫࢡࡢᐦᗘ (kg/m3)
tdisc㸸ࢹ࢕ࢫࢡࡢཌࡉ (m)
J pdisc
D
32
S
4
out
Din4 U disc t disc
2 J disc mdisc 2
t disc 㸦2.13㸧
6
࣮ࣟࢱ඲యࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝ
࣮ࣟࢱ඲యࡢ᭷㝈せ⣲ࣔࢹࣝࡢ㉁㔞⾜ิ[Mr]ࡣ㸪ᅇ㌿㍈ࡢ㉁㔞⾜ิ[Mshaft]ࡢᙜヱ⠇Ⅼ㒊ศ
࡟ࢹ࢕ࢫࢡࡢ㉁㔞⾜ิ[Mdisc]ࢆ㊊ࡋྜࢃࡏࡿࡇ࡜࡛᏶ᡂࡍࡿ㸬࣮ࣟࢱ඲యࡢ᭷㝈せ⣲ࣔࢹࣝ
ࡢ๛ᛶ⾜ิ[Kr]ࡣ㸪ᅇ㌿㍈ࡢ๛ᛶ⾜ิ[Kshaft]࡜ྠ୍࡛࠶ࡿ㸬ࡲࡓ㸪࣮ࣟࢱ඲యࡢࢪࣕ࢖ࣟ⾜
ิ[Gr]ࡣ㸪㍈ࡢࢪࣕ࢖ࣟ⾜ิ[Gshaft]ࡢᙜヱ⠇Ⅼ㒊ศ࡟ࢹ࢕ࢫࢡࡢࢪࣕ࢖ࣟ⾜ิ[Gdisc]ࢆ㊊ࡋྜ
ࢃࡏࡿࡇ࡜࡛᏶ᡂࡍࡿ㸬
ࡡࡌࡾ᣺ືࡢゎᯒࣔࢹࣝ
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢ⮬⏤ᗘ
ࡡࡌࡾ᣺ືࡢゎᯒࣔࢹ࡛ࣝࡣ㸪 z ㍈࿘ࡾࡢᅇ㌿ゅ șz ࡢ 1 ⮬⏤ᗘࡢࡳࢆᑐ㇟࡜ࡍࡿ㸦ᅗ 2.1
ཧ↷㸧㸬
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢ㐠ື᪉⛬ᘧ
ῶ⾶ຊࡀస⏝ࡋ࡞࠸ሙྜࡢࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡣ㸪ᘧ㸦2.14㸧࡛⾲ࡉࢀࡿ㸬
17
>J @^T ` >K @^T ` ^T ` 㸦2.14㸧
p
z
t
z
[Jp]㸸࣮ࣟࢱࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ⾜ิ㸪ࢧ࢖ࢬ n×n㸪ᑐ⛠⾜ิ
[Kt]㸸࣮ࣟࢱࡢࡡࡌࡾ๛ᛶ⾜ิ㸪ࢧ࢖ࢬ n×n㸪ᑐ⛠⾜ิ
{șz}㸸ゅᗘኚ఩࣋ࢡࢺࣝ㸪ࢧ࢖ࢬ n×1㸪
^T z ` ^T z1 ,T z 2 , ...,T zn `T
{T}㸸ࢺࣝࢡ࣋ࢡࢺࣝ㸪ࢧ࢖ࢬ n×1㸪
^T ` ^T1 , T2 ,... , Tn `T 㸪T ࡣࢺࣝࢡ
(Nm)
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝ࡟࠾ࡅࡿᅇ㌿㍈ࡢࣔࢹࣝ໬
㍈せ⣲ࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ⾜ิ[Jpe]ࡣ㸪ࢧ࢖ࢬ 2×2 ࡢᑐ⛠⾜ิ࡛࠶ࡾ㸪[Jpe]ࡢせ⣲ࡣᘧ
㸦2.15㸧࡛ᐃ⩏ࡉࢀࡿ㸦ᑠᯇ, 2009㸧㸬
> @
J pe
ª UI pele
«
« 3
« UI pele
«¬ 6
UI pele º
6
»
UI pele »»
㸦2.15㸧
3 »¼
ȡ㸸ᅇ㌿㍈ࡢᐦᗘ (kg/m3)
Ipe㸸ᅇ㌿㍈せ⣲ࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ (m4)㸪 I pe
Sd e 4 32
le㸸ᅇ㌿㍈せ⣲ࡢ㛗ࡉ (m)
㍈せ⣲ࡢࡡࡌࡾ๛ᛶ⾜ิ[Kte]ࡣ㸪ࢧ࢖ࢬ 2×2 ࡢᑐ⛠⾜ิ࡛࠶ࡾ㸪[Kte]ࡢせ⣲ࡣᘧ㸦2.16㸧
࡛ᐃ⩏ࡉࢀࡿ㸦ᑠᯇ, 2009㸧㸬
>K te @
ª GI pe
«
« le
« GI pe
« l
e
¬
GI pe º
»
le »
㸦2.16㸧
GI pe »
le »¼
G㸸ᶓᙎᛶಀᩘ (N/m2)
ᅇ㌿㍈඲యࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ[Jpshaft]㸦ࢧ࢖ࢬ n×n㸧ࡣ㸪᭤ࡆ᣺ືࣔࢹࣝࡢ㉁㔞⾜ิ࡜ྠ
ᵝ࡟㸪1 ␒┠ࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ⾜ิ[Jpe1]࠿ࡽ n-1 ␒┠ࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ⾜ิ[Jpe(n-1) ]ࢆ㔜
18
ࡡྜࢃࡏࡿࡇ࡜࡟ࡼࡗ࡚ᵓᡂࡉࢀࡿ㸦ᅗ 2.2 ཧ↷㸧㸬ᅇ㌿㍈඲యࡢࡡࡌࡾ๛ᛶ⾜ิ[Kt]ࡶྠᵝ
࡟ࡋ࡚ᵓᡂࡉࢀࡿ㸬ࡇࡇ࡛㸪㔜ࡡྜࢃࡏ࡜ࡣ༢⣧࿴࡛࠶ࡾ㸪ࡤࡡࡢ⤖ྜ᪉ἲ࡛ࡣ୪ิ⤖ྜ
࡟┦ᙜࡍࡿ㸬
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝ࡟࠾ࡅࡿࢹ࢕ࢫࢡࡢࣔࢹࣝ໬
1 ࡘࡢࢹ࢕ࢫࢡࡢ᭷㝈せ⣲ࣔࢹࣝࡣࢧ࢖ࢬ 2×2 ࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ⾜ิ[Jpdisc]ࡢࡳ࡛ᐃ⩏
ࡉࢀࡿ㸬ᘧ㸦2.17㸧࡟♧ࡉࢀࡿࡼ࠺࡟㸪ᴟ័ᛶ࣮࣓ࣔࣥࢺ⾜ิ[Jpdisc]ࡢせ⣲ࡣᐦᗘ ȡdisc㸪ᅇ
㌿㍈࿘ࡾࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ Ipdisc㸪࠾ࡼࡧཌࡉ tdisc ࡢ㛵ᩘ࡜ࡋ࡚ᐃ⩏ࡉࢀࡿ㸬
>J @
pdisc
ª U disc I pdisctdisc
«
3
«
« U disc I pdisctdisc
6
¬«
Udisc I pdisctdisc º
»
㸦2.17㸧
Udisc I pdisctdisc »»
3
¼»
6
ȡdisc㸸ࢹ࢕ࢫࢡࡢᐦᗘ (kg/m3)
Ipdisc㸸ࢹ࢕ࢫࢡࡢᅇ㌿㍈࿘ࡾࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ (m4)㸪 I pdisc
4
S Dout
Din4 32
tdisc㸸ࢹ࢕ࢫࢡࡢཌࡉ (m)
࣮ࣟࢱ඲యࡢࡡࡌࡾ᣺ືゎᯒࣔࢹࣝ
࣮ࣟࢱ඲యࡢ᭷㝈せ⣲ࣔࢹࣝࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ⾜ิ[Jp]ࡣ㸪ᅇ㌿㍈ࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ
⾜ิ[Jpshaft]ࡢᙜヱ⠇Ⅼ㒊ศ࡟ࢹ࢕ࢫࢡࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ⾜ิ[Jpdisc]ࢆ㊊ࡋྜࢃࡏࡿࡇ࡜࡛
᏶ᡂࡍࡿ㸬࣮ࣟࢱ඲యࡢ᭷㝈せ⣲ࣔࢹࣝࡢࡡࡌࡾ๛ᛶ⾜ิࡣ㸪ᅇ㌿㍈ࡢࡡࡌࡾ๛ᛶ⾜ิ[Kt]
࡜ྠ୍࡛࠶ࡿ㸬
⤖ゝ
୰ᐇ෇ᙧ᩿㠃ᙧ≧ࡢᅇ㌿㍈ࢆᣢࡘ࣮ࣟࢱ࡟ࡘ࠸࡚㸪᭤ࡆ᣺ືゎᯒ࡜ࡡࡌࡾ᣺ືゎᯒࡢࡓ
ࡵࡢ㸪1 ḟඖᱱࡢ᭷㝈せ⣲ゎᯒࣔࢹࣝࡢࣔࢹࣝ໬᪉ἲ࡟ࡘ࠸࡚ᴫㄝࡋࡓ㸬
ཧ⪃ᩥ⊩
Friswell, M.I., Penny, J. E. T., Garvey, S. D. and Lees, A. W., Dynamics of rotating machines,
Cambridge University Press, (2010).
ᑠᯇ ᩗ἞, ᶵᲔᵓ㐀᣺ືᏛ MATLAB ࡟ࡼࡿ᭷㝈せ⣲ἲ࡜ᛂ⟅ゎᯒ, ᳃໭ฟ∧ (2009).
㛗Ụ ಙ㢧, ᚋ⸨ ▱ఙ, ࣮ࣟࢱࡢ᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ≉ᛶ⾜ิࡢᐇ㦂ྠᐃ㸦⌮ㄽ࡜ࢩ࣑ࣗࣞ
࣮ࢩࣙࣥ㸧, ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟 C ⦅, Vol.79, No.803 (2013), pp.2314-2328.
19
➨ ❶ ㍈┤ᚄኚ໬㒊ࡢ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ໬
⥴ゝ
᪤Ꮡࡢ 1 ḟඖᱱࡢ᭷㝈せ⣲ゎᯒࣔࢹࣝ㸦௨ୗ㸪1D FEM ࣔࢹࣝ࡜⾲グ㸧ࡢࣔࢹࣝ໬᪉ἲ࡛
ࡣ㸪㍈┤ᚄኚ໬㒊࡟࠾ࡅࡿ኱ᚄ㍈᩿㠃࡜ᑠᚄ㍈࡜ࡢᅛᐃᗘపୗࡀ⪃៖ࡉࢀ࡞࠸ࡓࡵ㸪ࡑࡢ
㒊ศࡢ᭤ࡆ๛ᛶࢆ㐣኱ホ౯ࡍࡿࡇ࡜࡟࡞ࡿ㸬1.2.1 ⠇࡛㏙࡭ࡓ 1D ࣔࢹࣝ࡟࠾ࡅࡿᅇ㌿㍈ࡢ㍈
┤ᚄኚ໬㒊ࡢ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ໬࡟㛵ࡍࡿ᪤Ꮡ◊✲ࡢ࠺ࡕ㸪㧗⏣ࡽ㸦㧗⏣௚, 2009㸧࡟ࡼࡗ
࡚ᥦ᱌ࡉࢀࡓ㸪ᅇ㌿㍈ࡢ㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ㸪
ᙎᛶኚᙧ┦ᙜࡢ᭤ࡆ๛ᛶࢆᱱࡢ᭤ࡆ๛ᛶ࡟┤ิ⤖ྜࡍࡿ᪉ἲࢆᇶ࡟ࡋ࡚㸪1 ḟඖᱱࡢ᭷㝈せ
⣲ἲࡀ㐺⏝ࡉࢀࡓ㍈᣺ືゎᯒ⎔ቃ࡟㐺⏝ࡀᐜ᫆࡞᪉ἲࢆᥦ᱌ࡍࡿ㸬
3.2 ⠇࡛ࡣ㸪ᚑ᮶◊✲࡛ࡣ♧ࡉࢀ࡚࠸࡞࠸㍈┤ᚄẚ ȕ ࡟ࡘ࠸࡚㸪ᙎᛶኚᙧࢆ⪃៖ࡍ࡭ࡁ኱
ᚄ㒊ࡢᖜᚄẚ h/d0 ࡢ⠊ᅖࢆ᳨ウࡍࡿ㸬
3.3 ⠇࡛ࡣᥦ᱌ࡍࡿࣔࢹࣝ໬᪉ἲ࡟ࡘ࠸࡚㏙࡭ࡿ㸬ࡲࡎ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࡢ
➼౯᭤ࡆ๛ᛶ࡟ࡘ࠸᳨࡚ウࡋ㸪ḟ࡟㸪ᚑ᮶◊✲࡛♧ࡉࢀ࡚࠸ࡿ㸪ᅇ㌿ࡤࡡࡢ┤ิ⤖ྜ࡟ࡼ
ࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝࢆ㸪
1D FEM ࣔࢹࣝࡢሙྜ࡟ࡘ࠸᳨࡚ウࡍࡿ㸬ࡑࡋ࡚㸪ᅇ㌿ࡤࡡࡢ୪ิ⤖ྜ࡟ࡼࡗ࡚㍈┤ᚄኚ໬
㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝࢆᥦ᱌ࡍࡿ㸬᭱ᚋ࡟㸪
➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿ㍈ᚄ࡟ࡘ࠸᳨࡚ウࡍࡿ㸬
3.4 ⠇࡛ࡣ㸪ᥦ᱌᪉ἲࡢጇᙜᛶࡢ᳨ドࢆ⾜࠺㸬ࡲࡎ㸪ཧ↷࡜ࡍࡿ 3 ḟඖ᭷㝈せ⣲ࣔࢹࣝ㸦௨
ୗ㸪3D FEM ࣔࢹࣝ࡜⾲グ㸧ࡢ⢭ᗘࢆୖࡆࡿ᪉ἲࢆ㏙࡭ࡿ㸬ḟ࡟㸪᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘ
࡟ࡘ࠸࡚㸪ᥦ᱌᪉ἲࢆ㐺⏝ࡋࡓ 1D FEM ࣔࢹࣝ࡟ࡼࡿゎᯒ⤖ᯝ࡜㸪ཧ↷࡜࡞ࡿ 3D FEM ࣔ
ࢹࣝ࡟ࡼࡿゎᯒ⤖ᯝࡢẚ㍑ࢆ⾜࠺㸬ࡑࡋ࡚㸪㟼Ⲵ㔜࡟ࡼࡿ᭤ࡆኚᙧ࡟ࡘ࠸࡚㸪3D FEM ࣔࢹ
ࣝ࡟ᑐࡍࡿᥦ᱌᪉ἲࢆ㐺⏝ࡋࡓ 1D FEM ࣔࢹࣝࡢゎᯒ⢭ᗘࢆホ౯ࡍࡿ㸬ࡉࡽ࡟㸪᭤ࡆ࣮ࣔ
ࢻࡢᅛ᭷᣺ືᩘࡢᐇ㦂⤖ᯝࢆཧ↷್࡜ࡋ࡚ᥦ᱌᪉ἲࡢጇᙜᛶࢆホ౯ࡍࡿ㸬᭱ᚋ࡟㸪ලయ౛
࡜ࡋ࡚ᐇᶵ┦ᙜࡢゎᯒࣔࢹࣝ࡟ࡘ࠸࡚㸪ᥦ᱌᪉ἲࢆ㐺⏝ࡋࡓሙྜ࡜㐺⏝ࡋ࡞࠸ሙྜࡢ᭤ࡆ
࣮ࣔࢻࡢᅛ᭷᣺ືᩘࢆẚ㍑ࡋ࡚㸪࡝ࡢ⛬ᗘࡢᕪ␗ࡀ⏕ࡌࡿ࠿ࢆ♧ࡍ㸬
㐺⏝⠊ᅖࡢ᳨ウ
ⴭ⪅ࡀᥦ᱌ࡍࡿ㍈┤ᚄኚ໬㒊ࡢ᭤ࡆ๛ᛶࡢ㧗⢭ᗘ໬ἲࡣ㸪᪤Ꮡ◊✲࡛࠶ࡿ㧗⏣ࡽࡢ᪉ἲ
ࢆᇶ࡟ࡋࡓࡶࡢ࡛࠶ࡿࡓࡵ㸪㐺⏝࡛ࡁࡿ㍈┤ᚄẚ ȕ ࡜኱ᚄ㒊ࡢᖜᚄẚ h/d0 ࡢ⠊ᅖࡣ 1.2.1.2
⠇࡛♧ࡋࡓ ȕ=0.3175 ࡢሙྜࡣ h/d0 • 0.2 ࡛࠶ࡿ㸬᪤Ꮡ◊✲࡛ࡣ㸪ȕ ࡀ 0.3175 ࡜␗࡞ࡿሙྜࡢ
ᘧ㸦1.4㸧 ࡀᡂࡾ❧ࡘ h/d0 ࡢ⠊ᅖ࡟ࡘ࠸࡚ࡣ᳨ウࡉࢀ࡚࠸࡞࠸ࡓࡵ㸪ᅗ 1.4 ࡢ 3 ḟඖ᭷㝈せ
⣲ࣔࢹࣝࢆ⏝࠸࡚㸪㍈┤ᚄẚ ȕ=0.3175㸪ȕ=0.5650㸪࠾ࡼࡧ ȕ=0.7874 ࡢሙྜ࡟ࡘ࠸᳨࡚ウࢆ
⾜ࡗࡓ㸬
ᅗ 1.3 ࡢ 3D FEM ࣔࢹࣝࡢᑍἲࡣ d0=25.4mm㸪l2=12.7mm ࡜ࡋ㸪ᮦᩱ≉ᛶࡣࣖࣥࢢ⋡
E=2.050×1011 N/m2㸪ᶓᙎᛶಀᩘ G=7.880×1010 N/m2㸪࣏࢔ࢯࣥẚ Ȟ=0.3008 ࡜ࡋࡓ㸬
3D FEM ࣔࢹࣝ࡟ࡼࡿ㸪ᘧ㸦1.2㸧࡟࠾ࡅࡿ Į ࡜ h/d0 ࡢ㛵ಀࡢ⟬ฟᡭ㡰ࡣḟࡢ㏻ࡾ࡛࠶ࡿ㸬
20
ᅗ 1.3 ࡟࠾࠸࡚ᗙᶆཎⅬࢆ A-A ᩿㠃ࡢ㍈୰ᚰ࡟࠾ࡁ㸪x ㍈࿘ࡾࡢ᭤ࡆ࣮࣓ࣔࣥࢺ M ࡀᑠ
ᚄ㍈ࡢඛ➃ࡢᗙᶆ(0, 0, l2)࡟స⏝ࡋࡓ࡜ࡁࡢ yz ᖹ㠃࡟࠾ࡅࡿᑠᚄ㍈ࡢࡓࢃࡳ᭤⥺ࢆồࡵ࡚㸪
ᘧ㸦3.1㸧ࡢ 3 ḟᘧ࡛㏆ఝࡍࡿ㸬
y z c1 z 3 c2 z 2 c3 z , 0 d z d l2 㸦3.1㸧
c1, c2, c3: ࡓࢃࡳ᭤⥺ࡢಀᩘ
A-A ᩿㠃ࡢഴࡁゅ ȥ ࡣ㸪ᗙᶆཎⅬ࡛ࡢഴࡁ dy(0)/dz=c3 ࠿ࡽ
\
tan 1 c 3 㸦3.2㸧
࡛୚࠼ࡽࢀࡿ㸬ࡼࡗ࡚㸪኱ᚄ㒊ࡢ㍈᪉ྥ㛗ࡉ㸦෇ᯈࡢཌࡉ㸧h ࢆኚ࠼࡚ୗグࡢ (a)࠿ࡽ(d)ࡢ
ᡭ㡰ࢆᐇ᪋ࡍࡿࡇ࡜࡛ 1 ࡘࡢ ȕ ࡟ࡘ࠸࡚ Į ࡜ h/d0 ࡢ㛵ಀࢆồࡵࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
(a)
ᅗ 1.3 ࡢ 3D FEM ࣔࢹ࡛ࣝᑠᚄ㍈ࡢඛ➃࡟᭤ࡆ࣮࣓ࣔࣥࢺ M ࢆ୚࠼࡚ᑠᚄ㍈ࡢࡓࢃ
ࡳ᭤⥺ࢆồࡵࡿ㸬
(b)
ᑠᚄ㍈ࡢࡓࢃࡳ᭤⥺ࢆᘧ㸦3.1㸧ࡢ 3 ḟᘧ࡛᭤⥺㐺ྜࡋ࡚ཎⅬ࡛ࡢഴࡁ c3 ࢆồࡵࡿ㸬
(c)
ᘧ㸦3.2㸧࡟ࡼࡗ࡚ A-A ᩿㠃ࡢഴࡁゅ ȥ ࢆồࡵࡿ㸬
(d)
ᘧ㸦1.2㸧࡟ࡼࡗ࡚ Į ࢆồࡵࡿ㸬
ᅗ 3.1㸪ᅗ 3.2㸪࠾ࡼࡧᅗ 3.3 ࡣࡑࢀࡒࢀ㸪㍈┤ᚄẚ ȕ=0.3175㸪ȕ=0.5650㸪࠾ࡼࡧ ȕ=0.7874
ࡢሙྜ࡟ࡘ࠸࡚ Į ࡜ h/d0 ࡢ㛵ಀࢆồࡵࡓ⤖ᯝ࡛࠶ࡿ㸬ࡇࢀࡽࡢᅗ࡟ࡣᘧ㸦1.3㸧࡜ᘧ㸦1.4㸧
ࡢ್ࢆࡑࢀࡒࢀ㟷Ⰽ࡜⥳Ⰽࡢᐇ⥺࡛♧ࡋࡓ㸬ᘧ㸦1.4㸧ࡀᡂࡾ❧ࡘ h/d0 ࡢ⠊ᅖࡣ㸪࠸ࡎࢀࡢ
ȕ ࡢሙྜ࡟ࡶ㸪࠾ࡼࡑ h/d0 • 0.2 ࡛࠶ࡿ㸬ࡲࡓ㸪ȕ ࡀ኱ࡁࡃ࡞ࡿ⛬㸪ᘧ㸦1.4㸧࡟ࡼࡿ㏆ఝ⢭
ᗘࡀపୗࡋ࡚࠸ࡿ㸬
21
Equaation (A2.3)
3D F
FEM
Equaation (A2.2)
0.2
Fig. 3.1
1
Relationsship between
n Į and h/d0 (ȕ= 0.3175)
Equattion (A2.3)
3D FE
EM
Equattion (A2.2)
0.2
Fig. 3.2
2
Relationsship between
n Į and h/d0 (ȕ= 0.5650)
22
Equaation (A2.3)
3D F
FEM
Equaation (A2.2)
0.2
Fig. 3.3
3
Relationsship between
n Į and h/d0 (ȕ= 0.7874)
௨ୖ
ୖࡢ᳨ウࡼࡾ㸪ḟ⠇࡛ᥦ
ᥦ᱌ࡍࡿ㍈┤
┤ᚄኚ໬㒊ࡢࣔࢹࣝ໬ᡭἲ
ἲࡢ㐺⏝⠊ᅖ
ᅖࡣ㸪ȕ ࡟ࡼࡽࡎ㸪
࠾ࡼࡑ
ࡑ h/d0 • 0.2 ࡛࠶ࡿࡇ࡜ࡀ᫂ࡽ࠿࡟ ࡞ࡗࡓ㸬
㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧ࡟ࡼ
ࡼࡿぢ࠿ࡅࡢ
ࡢ㍈ࡢ᭤ࡆ๛
๛ᛶపୗࢆ⪃
⪃៖ࡋࡓࣔࢹ
ࢹࣝ໬
᪉ἲ
⠇࡛ࡣ㸪୍⯡
⯡ⓗ࡞ 1 ḟඖ
ඖᱱࡢ᭷㝈せ
せ⣲ࣔࢹࣝ㸦
㸦1D FEM ࣔࢹ
ࢹࣝ㸧ࢆ⏝࠸
࠸ࡓ㍈᣺ືゎ
ゎᯒࢯ
ᮏ⠇
ࣇࢺ࢙࢘࢔࡟㐺⏝
⏝ྍ⬟࡞㸪㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢࡳ࠿
࠿ࡅࡢ㍈ࡢ᭤
᭤ࡆ๛ᛶపୗ
ୗࢆ⪃៖ࡋࡓ
ࡓࣔࢹ
᪉ἲ࡟ࡘ࠸࡚
࡚㏙࡭ࡿ㸬
ࣝ໬᪉
᩿㠃
㠃ࡀ୍ᵝ࡞㍈
㍈せ⣲࡛ࡣ㸪࡝ࡢ㍈せ⣲
⣲࡟࠾࠸࡚ࡶ
ࡶ㸪ࡑࡢ୧➃
➃࡟࠾ࡅࡿ᭤
᭤ࡆ๛ᛶࡣ➼
➼ࡋ࠸
ࡇ࡜ࡀ
ࡀᮃࡲࡋ࠸㸬
㸬ࡋ࠿ࡋ㸪㍈
㍈┤ᚄኚ໬㒊
㒊ࡢせ⣲࡟࠾
࠾࠸࡚ࡣ㸪᭤
᭤ࡆ๛ᛶࡢప
పୗࡀ࠶ࡿࡓ
ࡓࡵ㸪
ࡑࡢ୧
୧➃࡟࠾ࡅࡿ
ࡿ᭤ࡆ๛ᛶࡀ
ࡀ␗࡞ࡗ࡚ࡋ
ࡋࡲ࠺㸬ࡑࡇ࡛㸪ᑠᚄഃ࡟
࡟᭷㝈㛗ࡉࡢ
ࡢ㍈せ⣲ࢆタ
タࡅ㸪
ࡑࡢせ
せ⣲ࡢ᭤ࡆ๛
๛ᛶࢆ㐺ษ࡟
࡟タᐃࡍࡿࡇ
ࡇ࡜࡟ࡼࡾ㸪㍈┤ᚄኚ໬
໬㒊ࡢࡳ࠿ࡅ
ࡅࡢ㍈ࡢ᭤ࡆ
ࡆ๛ᛶ
పୗࢆ⪃៖࡛ࡁࡿ
ࡿࣔࢹࣝ໬᪉
᪉ἲࢆᥦ᱌ࡍ
ࡍࡿ㸬
ኚ໬㒊᩿㠃ࡢ
ࡢᙎᛶኚᙧࡢ
ࡢ➼౯᭤ࡆ๛ᛶ
㍈┤ᚄኚ
ᑠᚄ
ᚄ㍈ࡢඛ➃࡟
࡟స⏝ࡍࡿ᭤
᭤ࡆ࣮࣓ࣔࣥ
ࣥࢺ M ࡟ࡼࡗ࡚⏕ࡌࡿ㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢഴࡁ
ࡁゅ ȥ
㸦ᅗ 1.3 ཧ↷㸧ࢆ
ࢆ㸪∦ᣢࡕᱱ
ᱱࡢࡓࢃࡳゅ ࡜➼౯࡜࠾࠸࡚㸪㍈┤ᚄ
ᚄኚ໬㒊᩿㠃
㠃ࡢᙎᛶኚᙧ
ᙧࡢ➼
ࡆ๛ᛶࢆồࡵ
ࡵࡿ㸬
౯᭤ࡆ
᭤ࡆ
ࡆ࣮࣓ࣔࣥࢺ M ࡜㍈┤ᚄ
ᚄኚ໬㒊᩿㠃
㠃ࡢࡓࢃࡳゅ
ゅ ȥ ࡢ㛵ಀࡣ
ࡣᘧ㸦1.2㸧࡛
࡛୚࠼ࡽࢀࡿ
ࡿ㸬
23
ᅗ 3.4 ࡟♧ࡉࢀ
ࢀࡿ㛗ࡉ l㸪ࣖࣥࢢ⋡ E㸪
㸪᩿㠃 2 ḟࣔ
࣮࣓ࣔࣥࢺ Id ࡢ୍ᵝ᩿㠃
㠃ࡢ∦ᣢࡕᱱ
ᱱࡀඛ
ࣥࢺ M ࢆཷࡅࡓ࡜ࡁࡢඛ
ඛ➃ࡢࡓࢃࡳ
ࡳゅ ȥ ࡣ㸪ᘧ
ᘧ㸦3.3㸧࡛୚
୚࠼ࡽࢀࡿ㸦୰ἑ
➃࡟᭤ࡆ࣮࣓ࣔࣥ
௚, 19973a㸧㸬
M
ȥ
E
E, Id
l
Fig. 3.4 Slope of canntilever beam
m due to bend
ding moment
\
Ml
㸦3.3㸧
EI d
ᘧ㸦11.2㸧࡜ᘧ㸦3.3㸧࠿ࡽ㸪㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧࡢ➼౯᭤ࡆ๛ᛶ EIId ࢆ⾲ࡍᘧ㸦
㸦3.4㸧
ࡀᚓࡽࢀࡿ㸬
EI d
EDh 3 l 㸦3.4㸧
ᅇ㌿ࡤࡡ
ࡡࡢ┤ิ⤖ྜ
ྜ࡟ࡼࡗ࡚㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧ࡟ࡼࡿ
ࡿぢ࠿ࡅࡢ㍈
㍈ࡢ᭤
ࡆ๛ᛶ
ᛶపୗࢆ⪃៖
៖ࡋࡓࣔࢹࣝ
ࣝ
1D
D FEM ࣔࢹࣝ
ࣝ࡟࠾࠸࡚㸪㍈┤ᚄኚ໬ 㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧ࡟ࡼࡿ
ࡿࡳ࠿ࡅࡢ㍈
㍈ࡢ᭤ࡆ๛ᛶ
ᛶపୗ
ࢆ⪃៖
៖ࡋࡓࣔࢹࣝ
ࣝࢆᅗ 3.5 ࡟♧ࡍ㸬
࡟
A
h
ȭd 0
I1
1
F
Fig. 3.5
I”=I1’+I2’
I2
3
2
element 1
l2
A
ȭd
element 2
1D beam
b
model for bending iin consideratiion for elasticcity of stepped
ed cross-sectio
on
24
3 ࡘࡢ⠇Ⅼ࡛ 2 ࡘࡢ㍈せ⣲ࢆᐃ⩏ࡋ㸪኱ᚄ㍈ྑ➃ࡢ⠇Ⅼ 2 ࡢ᭤ࡆ๛ᛶ EI1’࡜ᑠᚄ㍈ࡢᕥ➃
ࡢ⠇Ⅼ 2 ࡢ᭤ࡆ๛ᛶ EI2’ࢆࡑࢀࡒࢀᘧ㸦3.5㸧࡜ᘧ㸦3.6㸧ࡢ┤ิ⤖ྜ࡟ࡼࡗ࡚ồࡵࡿ㸬
EI 1c
E
I1 I d
㸦3.5㸧
I1 I d
E
I2Id
㸦3.6㸧
I2 Id
EI 2c
ḟ࡟㸪኱ᚄ㍈࡜ᑠᚄ㍈ࡢ⤖ྜ㒊㸦⠇Ⅼ 2㸧ࡢ᭤ࡆ๛ᛶࢆ㸪㏻ᖖࡢ㔜ࡡྜࢃࡏ㸪ࡍ࡞ࢃࡕ୪
ิ⤖ྜ࡟ࡼࡗ࡚ồࡵࡿ㸬
EI cc
EI 1c EI 2c
§ I I
I I
E ¨¨ 1 d 2 d
© I1 I d I 2 I d
·
¸ 㸦3.7㸧
¸
¹
ᘧ㸦3.5㸧㸪㸦3.6㸧㸪㸦3.7㸧ࢆᅗ 3.6(a)࡜ᅗ 3.6(b)ࢆ⏝࠸࡚ㄝ᫂ࡍࡿ㸬
ᅗ 3.6(a)ࡣ㸪2 ࡘࡢ㍈せ⣲࡟ࡘ࠸࡚ 1 ࡘࡢᅇ㌿⮬⏤ᗘ࡟╔┠ࡋࡓ័ᛶ࣮࣓ࣔࣥࢺ࡜ᅇ㌿ࡤ
ࡡࡢ⤖ྜࢆᶍᘧⓗ࡟♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬
ᘧ㸦2.1㸧ࡢ㐠ື᪉⛬ᘧ࡟࠾ࡅࡿኚ఩࣋ࢡࢺࣝ{X}ࡢ x ㍈࿘ࡾࡢᅇ㌿ゅ șx ࡢࡳࢆ⪃࠼ࡿ࡜㸪
ᘧ㸦2.7㸧ࡼࡾ㸪⠇Ⅼ 1 ࡜⠇Ⅼ 2 ࡛ᐃ⩏ࡉࢀࡿせ⣲ 1 ࡢᅇ㌿ࡤࡡ k1 ࡜⠇Ⅼ 2 ࡜⠇Ⅼ 3 ࡛ᐃ⩏
ࡉࢀࡿせ⣲ 2 ࡢᅇ㌿ࡤࡡ k2 ࡣࡑࢀࡒࢀ㸪k1
EI 1 4 W ^1 W ˜ h` ࡜ k 2
EI 2 4 W ^1 W ˜ l 2 `
࡛୚࠼ࡽࢀ㸪ࡇࢀࡽࡢᅇ㌿ࡤࡡࡀᅗ 2.2 ࡟♧ࡉࢀࡿᵝ࡟ 2 ࡘࡢせ⣲ࡀ㔜」ࡍࡿ⠇Ⅼ 2 ࡛ࡣ㔜
ࡡྜࢃࡏ㸪ࡍ࡞ࢃࡕ୪ิ⤖ྜࡉࢀ࡚ k1+k2 ࡜࡞ࡿ㸬せ⣲ 1 ࡜せ⣲ 2 ࡢ័ᛶ࣮࣓ࣔࣥࢺ m1 ࡜
m2 ࡟ࡘ࠸࡚ࡶᅇ㌿ࡤࡡ࡜ྠᵝ࡟㔜ࡡྜࢃࡏࡽࢀ࡚ 2 ࡘࡢせ⣲ࡀ㔜」ࡍࡿ⠇Ⅼ 2 ࡛ࡣ m1+m2
࡜࡞ࡿ㸬ࡲࡓ㸪୍ᵝ᩿㠃ࡢ 1 ࡘࡢ㍈せ⣲ࡢ୧➃࡟࠾࠸࡚㉁㔞㸪័ᛶ࣮࣓ࣔࣥࢺ㸪୪㐍ࡤࡡ㸪
ᅇ㌿ࡤࡡࡣྠ୍ࡢ್࡜࡞ࡿ㸬ణࡋ㸪ᅗ 3.6(a)࡛ࡣ㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅ
ࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࡣ⪃៖ࡉࢀ࡚࠸࡞࠸㸬
25
m1
m1
k1
k1
node 2
node 1
k2
k2
node 2
=
k1 +k2
k2
m1
m1 +m2
k1
k2
k1
node 1
Fig. 3.6 (a))
node
n
3
m2
m2
m2
+
node 2
node
n
3
Assemblin
ng rotational springs for tw
wo adjacent elements
e
in 1D
D FEM modeel
- Parallel connnection of eaach rotational springs for ttwo beam eleements
᪉㸪ᅗ 3.6(b)ࡣ㸪㍈┤ᚄ
ᚄኚ໬㒊ࡢ⠇ Ⅼ 2 ࡟࠾ࡅࡿ᩿㠃ࡢᙎᛶ
ᛶኚᙧ࡟ࡼࡿ
ࡿࡳ࠿ࡅࡢ㍈
㍈ࡢ᭤
୍᪉
ࡆ๛ ᛶపୗࢆ㸪 ᩿㠃ࡢᙎᛶ
ᛶኚᙧࢆ⾲ࡍ
ࡍᅇ㌿ࡤࡡ kd ࢆ⏝࠸࡚
࡚㸪 k1 ࡜ kd ࡢ┤ิ⤖ྜ
ྜࡤࡡ
k1 k d / k1 k d ࡜ k2 ࡜ kd ࡢ┤
┤ิ⤖ྜࡤࡡ
ࡡ k 2 k d / k 2 k d ࢆ୪ิ⤖
⤖ྜࡋࡓࣔࢹ
ࢹ࡛ࣝ⾲ࡋࡓ
ࡓࡶࡢ
࡛࠶ࡾ㸪⠇Ⅼ 2 ࡟࠾ࡅࡿᅇ㌿
࡟
㌿ࡤࡡࡣᅗ 3..6(b)୰࡟♧ࡉ
ࡉࢀࡿᵝ࡟ k1 k d / k1 k d k 2 k d / k 2 k d ࡜࡞ࡿ㸬
26
m1
m1
k1
k1
node 2
de 1
nod
+
kd
kd
noode 2
k2
m2
m2
+
k2
node 2
=
node 3
kk
k1kd
2 d
k1 kd k2 kd
kd
k2
m1
k1
nod
de 1
k2
node 2
m2
kd
m1 +m2
k1
node 3
Fig. 3.6 (b)) Assemblin
ng rotational springs for tw
wo adjacent elements
e
in 1D
D FEM modeel
- Parrallel connecttion of each sseries-conneccted rotational springs for ttwo beam eleements
27
ᘧ㸦3.7㸧࡟࠾࠸࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⾲ࡍ᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ Id ࡣᘧ㸦3.4㸧
࡛୚࠼ࡽࢀࡿࡀ㸪ࡇࡢࡲࡲ࡛ࡣ㍈せ⣲㛗 l ࡟ࡼࡗ࡚ Id ࡀኚ໬ࡋ࡚ࡋࡲ࠺ࡓࡵ㸪ጇᙜ࡞ l ࡢ್
ࢆ᳨ウࡍࡿ㸬ᘧ㸦3.4㸧࡟ᘧ㸦1.4㸧ࢆ௦ධࡍࡿ࡜෇ᯈࡢཌࡉ h ࡀᾘཤࡉࢀ㸪ᘧ㸦3.8㸧࡟♧ࡉ
ࢀࡿᵝ࡟᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ Id ࡣᑠᚄ㍈ࡢせ⣲ࡢ┤ᚄ d ࡜㛗ࡉ l ࡢࡳ࡛⾲ࡉࢀࡿ㸬
S
Id
18
d 3 l 㸦3.8㸧
᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ Id ࢆᑠᚄ㍈ࡢ᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ I2 ࡜➼౯࡜࠾ࡃ࡜㸪୰ᐇ෇ᙧ᩿㠃㍈
࡛ࡣ
Id
I2
S
64
d 4 㸦3.9㸧
࡜࡞ࡾ㸪ᘧ㸦3.8㸧࡜ᘧ㸦3.9㸧ࡼࡾ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧศ࡜➼౯࡞᭤ࡆ๛ᛶࢆ୚
࠼ࡿᑠᚄ㍈ࡢ㛗ࡉࢆ⾲ࡍᘧ㸦3.10㸧ࡢ᮲௳ࡀᚓࡽࢀࡿ㸬
l
9
d 㸦3.10㸧
32
ᘧ㸦3.10㸧ࡣ㸪᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ㍈┤ᚄኚ໬㒊᩿㠃ࡢഴࡁゅ࡜ᑠᚄ㍈ࡢࡓࢃࡳゅࡀ➼౯
࡜࡞ࡿ㸪ᑠᚄ㍈ࡢ㛗ࡉࢆ♧ࡋ࡚࠸ࡿ㸬
ᘧ㸦3.10㸧࡛୚࠼ࡽࢀࡿ㛗ࡉࢆᣢࡘ┤ᚄ d ࡢᑠᚄ㍈ࡢせ⣲ࡢ᭤ࡆ๛ᛶ EI2 ࡀ㍈┤ᚄኚ໬㒊
᩿㠃ࡢ᭤ࡆ๛ᛶ EId ࡜➼౯࡛࠶ࡿࡓࡵ㸪
EI d
EI 2
㸦3.11㸧
࡜࠾ࡃࡇ࡜ࡀ࡛ࡁ㸪᩿㠃 A-A ࡟࠾ࡅࡿ᭤ࡆ๛ᛶ EIA ࢆ⾲ࡍᘧ㸦3.7㸧ࡣ㸪ᘧ㸦3.12㸧࡜࡞ࡿ㸬
EI A
EI cc
§ I I
I I
E ¨¨ 1 2 2 2
© I1 I 2 I 2 I 2
·
¸¸
¹
§ I I
I ·
E ¨¨ 1 2 2 ¸¸ 㸦3.12㸧
© I1 I 2 2 ¹
ᘧ㸦3.12㸧࡟ࡼࡗ࡚㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⪃
៖ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࡀ㸪኱ᚄ㍈せ⣲㸪ᑠᚄ㍈せ⣲㸪ඹ࡟㍈せ⣲ࡢ୧➃࡛᭤ࡆ๛ᛶࡀ␗࡞ࡿ
㸪୍⯡ⓗ࡞ 1D FEM ࣔࢹࣝࡀ㐺⏝ࡉࢀࡓ㍈᣺ືゎᯒ⎔ቃ࡟ࡇࡢ᪉
ࡓࡵ㸦EI1EI1’㸪EI2’EI2㸧
ἲࢆ㐺⏝ࡍࡿࡇ࡜ࡣᐜ࡛᫆ࡣ࡞࠸㸬
28
ᅇ㌿ࡤࡡ
ࡡࡢ୪ิ⤖ྜ
ྜ࡟ࡼࡗ࡚㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧ࡟ࡼࡿ
ࡿࡳ࠿ࡅࡢ㍈
㍈ࡢ᭤
ࡆ๛ᛶ
ᛶపୗࢆ⪃៖
៖ࡋࡓࣔࢹࣝ
ࣝ
୍⯡
⯡ⓗ࡞ 1D FEM
F
ࣔࢹࣝ
࡛ࣝ⏝࠸ࡽࢀ ࡚࠸ࡿᅇ㌿ࡤࡡࡢ୪ิ⤖
⤖ྜ࡟ࡼࡗ࡚
࡚㸪㍈┤ᚄኚ
ኚ໬㒊
᩿㠃ࡢ
ࡢᙎᛶኚᙧ࡟
࡟ࡼࡿࡳ࠿ࡅ
ࡅࡢ㍈ࡢ᭤ࡆ
ࡆ๛ᛶపୗࢆ
ࢆ⪃៖ࡍࡿࡇ
ࡇ࡜ࡀ࡛ࡁࡿ
ࡿࣔࢹࣝ໬᪉
᪉ἲࢆ
ᥦ᱌ࡍ
ࡍࡿ㸬
ᅗ 3.7 ࡣ㸪ࡇࡇ
ࡇ࡛ᥦ᱌ࡍࡿ
ࡿ㍈┤ᚄኚ໬
໬㒊ࡢࣔࢹࣝ
ࣝ໬᪉ἲࢆ♧ࡋࡓࡶࡢ࡛࠶
࠶ࡿ㸬
l
A
h
l 2 ’ = l 2 -l
ȭd 0
I2
1
3
2
element
A
4
ȭd
Ieq
I1
Fig. 3.7
9
d
32
element
1D beam model
m
in conssideration forr elasticity of stepped crosss-section
using
u
the methhod proposed
d by the autho
or
ࡲࡎ
ࡎ㸪㛗ࡉ 9d/32 ࡢ㍈せ⣲
⣲ࢆ㍈┤ᚄኚ
ኚ໬㒊ࡢᑠᚄ
ᚄ㒊࡟タࡅࡿ㸬
ḟ࡟
࡟㸪ᘧ㸦3.122㸧࡛♧ࡉࢀ
ࢀࡿ᩿㠃 A-A
A ࡟࠾ࡅࡿ᭤
᭤ࡆ๛ᛶ EIA ࢆ㸪୍⯡ⓗ࡞
ࢆ
࡞ 1D FEM ࣔࢹࣝ
ࣔ
࡛⏝࠸
࠸ࡽࢀࡿ㔜ࡡ
ࡡྜࢃࡏ㸪ࡍ
ࡍ࡞ࢃࡕ୪ิ
ิ⤖ྜ࡛⾲ࡍ
ࡍࡇ࡜ࢆ⪃࠼
࠼ࡿ㸬ᅗ 3.77 ࡟࠾࠸࡚㸪㛗ࡉ
9d/322 ࡢᑠᚄ㍈せ
せ⣲ࡢ➼౯᭤
᭤ࡆ๛ᛶࢆ EIIeq ࡜࠾࠸࡚⠇
⠇Ⅼ 2 ࡜⠇Ⅼ
Ⅼ 3 ࡛ྛࠎࡢ
ࡢ᭤ࡆ๛ᛶࡀ
ࡀ୪ิ
⤖ྜࡉࢀ㸪⠇Ⅼ 2㸪3 ࡢ᭤ࡆ๛
๛ᛶࡀ┤ิ⤖
⤖ྜࡉࢀࡓࡶ
ࡶࡢࡀᘧ㸦3.1
13㸧࡛࠶ࡾ㸪ࡇࢀࡣᘧ㸦3.12
3 㸧
౯࡛࠶ࡿ㸬
࡜➼౯
EI A
E
I I ˜ I
I I I
1
1
eq
eq
2
2
I eq
I eq
㸦3.13㸧
ᘧ㸦3.13㸧࡟ࡘ
ࡘ࠸࡚ᅗ 3.8 ࢆ⏝࠸࡚ㄝ ᫂ࡍࡿ㸬
3 ࡢ㍈┤ᚄ
ᚄኚ໬㒊ࡢ⠇ Ⅼ 2 ࡟࠾ࡅࡿ᩿㠃ࡢᙎᛶ
ᛶኚᙧ࡟ࡼࡿ
ࡿࡳ࠿ࡅࡢ㍈
㍈ࡢ᭤
ᅗ 3.8 ࡣ㸪ᅗ 3.7
ᛶపୗࢆ㸪⠇
⠇Ⅼ 2 ࡜⠇Ⅼ
Ⅼ 3 ࡛ᐃ⩏ࡉ
ࡉࢀࡿ㛗ࡉ 9d/32 ࡢ㍈せ
せ⣲ࡢᅇ㌿ࡤ
ࡤࡡ keq ࢆ⏝࠸
࠸࡚㸪
ࡆ๛ᛶ
⠇Ⅼ 2 ࡜⠇Ⅼ 3 ࡢࢺ࣮ࢱࣝࡢ
ࡢ
ࡢᅇ㌿ࡤࡡ࡛
࡛⾲ࡋࡓࣔࢹ
ࢹ࡛ࣝ࠶ࡿ㸬ᅗ 3.8 ࡛ࡣ㸪
㸪㛗ࡉ 9d/32
2 ࡢ㍈
ࡢ័ᛶ࣮࣓ࣔ
࣓ࣥࢺࢆ md ࡛⾲ࡋ࡚࠸ ࡿ㸬⠇Ⅼ 2 ࡢᅇ㌿ࡤࡡࡣ
ࡢ
ࡣせ⣲ 1 ࡢᅇ
ᅇ㌿ࡤࡡ k1 ࡜㛗ࡉ
࡜
せ⣲ࡢ
9d/322 ࡢせ⣲ࡢᅇ
ᅇ㌿ࡤࡡ keq ࡀ୪ิ⤖ྜࡉ
ࡀ
ࡉࢀ࡚ k1+keqq ࡜࡞ࡾ㸪⠇
⠇Ⅼ 3 ࡢᅇ㌿
㌿ࡤࡡࡣ㛗ࡉ 9d/32
29
⣲ࡢᅇ㌿ࡤࡡ
ࡡ keq ࡜せ⣲
⣲ 2 ࡢᅇ㌿ࡤ ࡡ k2 ࡀ୪ิ
ิ⤖ྜࡉࢀ࡚ k2+keq ࡜࡞ࡿ
ࡿ㸬⠇Ⅼ 2 ࡜⠇Ⅼ
࡜
ࡢせ⣲
3 ࡢࢺ
ࢺ࣮ࢱࣝࡢᅇ
ᅇ㌿ࡤࡡࡣ㸪⠇Ⅼ 2 ࡢᅇ
ᅇ㌿ࡤࡡ k1+keq ࡜⠇Ⅼ 3 ࡢᅇ㌿ࡤࡡ k2+keq ࡀ┤ิ
ิ⤖ྜ
ࡉࢀ࡚
࡚(k1+keq)‚(k2+keq)/{( k1+keq)+( k2+keq) } ࡜࡞ࡿ㸬
1
1
eq
2
eq
2
k eq
node 1
k2
md +m2
m1+md
m1
keq
k1
keq
k1
Fig. 3.8
k eq
m2
k k ˜ k
k k k
k2
node 3
noode 2
node 4
Assembling
g rotational spprings for treee adjacent eleements in 1D FEM model
ᘧ㸦3.13㸧ࡀᘧ
㸦3.12㸧࡜➼౯
౯࡛࠶ࡿࡢ࡛ 㸪
᭤ࡆ๛ᛶࡢ
ࡢ➼౯᩿㠃 2 ḟ࣮࣓ࣔࣥ
ࣥࢺ Ieq ࡣᘧ㸦3.14
3 㸧
࡜࡞ࡿ㸬
2
I eq
4
2
2
3
I 1 I 2 I 1 I 1 7 I 1 I 2 6 I 1 I 2 2I 2
2I 1 I 2 4
㸦3.14㸧
ᅗ 3.8 ࡢ⠇Ⅼ 2 ࡜⠇Ⅼ 3 ࡛ᐃ⩏ࡉࢀࡿ
࡛
ࡿ㛗ࡉ 9d/32 ࡢ㍈せ⣲ࢆ㍈
㍈┤ᚄኚ໬㒊
㒊ࡢᑠᚄഃ࡟
࡟タࡅ
ࡑࡢ㒊ศࡢ᭤
᭤ࡆ๛ᛶࢆ EI
E eq ࡜࠾ࡃࡇ
ࡇ࡜࡛㸪᩿㠃
㠃 A-A ࡢᙎᛶ
ᛶኚᙧࢆ⪃៖
៖ࡋࡓ᭤ࡆ๛
๛ᛶࢆ
࡚㸪ࡑ
⾲⌧ࡍ
ࡍࡿࡇ࡜ࡀ࡛
࡛ࡁࡿ㸬ࡓࡔ
ࡔࡋ㸪ᘧ㸦33.14㸧ࡀᡂ❧
❧ࡍࡿࡢࡣ㸪ᘧ㸦1.4㸧ࡀ
ࡀᡂ❧ࡍࡿ࠾
࠾ࡼࡑ
h/ d0 • 0.2 ࡢ⠊ᅖ
ᅖ࡟࠾࠸࡚࡛࠶ࡿ㸬ࡲࡓ 㸪ᑠᚄ㍈ࡢ㛗
㛗ࡉ l2 ࡀ 9d/
d 32 ࡟㏆ࡃ࡞
࡞ࡿࡼ࠺࡞▷
▷࠸㍈
せ⣲ࢆ⏝࠸ࡊࡿࢆ
ࢆᚓ࡞࠸ࡼ࠺࡞ሙྜࡣ㐺
㐺⏝ᑐ㇟እ࡜
࡜࡞ࡿ㸬ࡋ࠿
࠿ࡋ㸪ࡇࢀࡽ
ࡽࡢไ⣙ࢆཷ
ཷࡅ࡞
≧ࡢ㍈࡟ࡘ࠸
࠸࡚ࡣ㸪᭤ࡆ
ࡆ᣺ືゎᯒࣔ
ࣔࢹࣝࡢࣔࢹ
ࢹࣝ໬⢭ᗘࢆ
ࢆྥୖࡉࡏࡿ
ࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
࠸ᙧ≧
᭤ࡆ๛ᛶ
ᛶࡢ➼౯㍈ᚄ
ᚄࡢ᳨ウ
ᮏ⠇
⠇࡛ࡣ㸪㍈┤
┤ᚄኚ໬㒊ࡢ
ࡢࡳ࠿ࡅࡢ㍈
㍈ࡢ᭤ࡆ๛ᛶ
ᛶపୗࢆᢕᥱ
ᥱࡋ᫆ࡃࡍࡿ
ࡿࡓࡵ࡟㸪➼
➼౯࡞
᭤ࡆ๛
๛ᛶࢆ୚࠼ࡿ
ࡿᑠᚄ㒊ࡢ㍈
㍈ᚄ deq ࢆᐃ
ᐃ⩏ࡋ㸪deq ࡀ㍈┤ᚄẚ ȕ (=
( d/d0)࡟ᑐࡋ
ࡋ࡚࡝ࡢࡼ࠺
࠺࡟ኚ
໬ࡍࡿ࠿ࢆ᳨ウࡍ
ࡍࡿ㸬
౯࡞᭤ࡆ๛ᛶ
ᛶࢆ୚࠼ࡿᑠ
ᑠᚄ㒊ࡢ㍈ᚄ ࢆ deq ࡜ࡋ࡚
࡚㸪ᘧ㸦3.14㸧࡛♧ࡉࢀࡿ
㸧
ࡿ᭤ࡆ๛ᛶࡢ
ࡢ➼౯
➼౯
30
ᘧ㸦3.9㸧ࡢ୰
୰ᐇ෇ᙧ᩿㠃
㠃㍈ࡢ᩿㠃 2 ḟ࣮࣓ࣔࣥ
ࣥࢺࡢᙧ࡛⾲ࡍ࡜㸪
᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ Ieq ࢆᘧ
ࡿ㸬
ᘧ㸦33.15㸧࡜࡞ࡿ
I eq
S
4
64
d eq 㸦3.15㸧
ᘧ㸦33.15㸧ࡼࡾ㸪➼౯࡞᭤ࡆ
ࡆ๛ᛶࢆ୚࠼
࠼ࡿᑠᚄ㒊ࡢ
ࡢ㍈ᚄ deq ࡣᘧ
ᘧ㸦3.16㸧࡛
࡛⾲ࡉࢀࡿ㸬
d eq
4
64
S
I eeq 㸦3.16㸧
ධࡋ࡚ Ieq ࢆ d0 ࡜ d ࡢ㛵
ᘧ㸦3.14㸧࡟ I1=ʌd04/64㸪 I2=ʌd4/64 ࢆ௦ධ
㛵ᩘ࡛⾲ࡋ㸪㍈
㍈┤ᚄẚ ȕ (=
( d/d0)
౯㍈ᚄẚ deqq/d ࡢ㛵ಀࢆ
ࢆồࡵࡿ࡜㸪 ᅗ 3.9 ࡢ᭤⥺
⥺ࡀᚓࡽࢀࡿ
ࡿ㸬ࡇࡢ㛵ಀ
ಀࡢ≉ᚩࡣ㸪㍈┤
࡜➼౯
ᚄẚ ȕ ࡀᑠࡉࡃ࡞
࡞ࡿ࡜➼౯㍈
㍈ᚄẚ deq/d ࡀ 0.8409 ࡟₞㏆ࡍࡿⅬ
࡟
࡛࠶ࡿ㸬ࡇࢀ
ࢀࡼࡾ㸪ᘧ㸦1.4㸧
㍈┤ᚄ d0 ࡀᑠ
ᑠᚄ㍈┤ᚄ d ࡟ᑐࡋ࡚࠸
࠸ࡃࡽ኱ࡁࡃ
ࡃ࡞ࡗ
ࡀᡂࡾ❧ࡘ h/d0 • 0.2 ࡢ⠊ᅖ࡛ࡣ㸪኱ᚄ㍈
㸪➼౯࡞᭤ࡆ
ࡆ๛ᛶ EIeq ࡣᑠᚄ㍈ࡢ᭤
ࡣ
᭤ࡆ๛ᛶ EI2 ࡢ 0.84094=0
0.4979 ಸࢆୗ
ୗᅇࡽ࡞࠸࡜
࡜ゝ࠼
࡚ࡶ㸪
ࡿ㸬
Fiig. 3.9
Relaationship betw
ween ȕ and deq
e /d
ୖ࡟ࡼࡾ㸪㛗
㛗ࡉ 9d/32 ࡢᑠᚄ㍈せ⣲
ࡢ
⣲ࡢ᭤ࡆ๛ᛶ
ᛶࢆ EIeq ࡜ࡍࡿ㸪࠶ࡿ࠸ࡣ
ࡣ㸪㛗ࡉ 9d/32 ࡢ
௨ୖ
ᑠᚄ㍈
㍈せ⣲ࡢ┤ᚄ
ᚄࢆᘧ㸦3.16
6㸧ࡢ➼౯࡞
࡞᭤ࡆ๛ᛶࢆ
ࢆ୚࠼ࡿᑠᚄ㒊ࡢ㍈ᚄ deeq ࡜ࡍࢀࡤ㸪୍⯡
ⓗ࡞୍
୍ḟඖᱱࣔࢹ
ࢹࣝࡀ㐺⏝ࡉ
ࡉࢀࡓ㍈᣺ື
ືゎᯒࢯࣇࢺ࢙࢘࢔࡛㸪㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚ
ᙧࢆ⪃
⪃៖ࡋࡓ㍈᣺
᣺ື㸦᭤ࡆ᣺
᣺ື㸧ࡢゎᯒ
ᯒࢆ⡆౽࡟ᐇ
ᐇ⌧ࡍࡿࡇ࡜
࡜ࡀ࡛ࡁࡿ㸬 ࡓࡔࡋ㸪ᮏ
ᮏᥦ᱌
31
᪉ἲ࡛ࡣ㛗ࡉ 9d/32 ࡢᑠᚄ㍈せ⣲࡛㍈┤ᚄኚ໬㒊ࡢࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⾲⌧ࡋ࡚
࠸ࡿࡓࡵ㸪➼౯㍈ᚄ deq ࢆ⏝࠸ࡓࣔࢹࣝ໬࡛ࡣ㍈ࡢ㉁㔞࡜㍈ࡢ᭷ຠࡏࢇ᩿㠃✚ࢆᐇ㝿ࡢ㍈ࡼ
ࡾᑠࡉࡃぢ✚ࡶࡿࡇ࡜࡟࡞ࡿ㸬ࡑࡇ࡛㸪➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿᑠᚄ㒊ࡢ㍈ᚄ deq ࢆ౑ࡗ࡚
㧗⢭ᗘ࡞ࣔࢹࣝ໬ࢆ⾜࠺࡟ࡣ㸪ᑠᚄ㍈ࡢ㉁㔞ࢆ⥔ᣢࡍࡿࡓࡵ࡟㛗ࡉ 9d/32 ࡢᑠᚄ㍈せ⣲ࡢ㒊
ศ࡟እᚄ d㸪ෆᚄ deq ࡢࢹ࢕ࢫࢡせ⣲ࢆ௜ຍࡋ㸪᭷ຠࡏࢇ᩿㠃✚ࢆ ț‚ʌdeq2/4 ࡛ࡣ࡞ࡃ ț‚ʌd2/4
࡜ࡍࢀࡤⰋ࠸㸦ᘧ㸦2.5㸧ཧ↷㸧㸬
ᥦ᱌᪉ἲࡢጇᙜᛶࡢ᳨ド
ᮏ⠇࡛ࡣ㸪ᥦ᱌ࡍࡿ㍈┤ᚄኚ໬㒊ࡢࣔࢹࣝ໬᪉ἲࢆ㐺⏝ࡋࡓ 1D FEM ࣔࢹ࡛ࣝồࡵࡓ㸪
⮬⏤ᨭᣢ≧ែࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘ࡜㟼Ⲵ㔜࡟ࡼࡿ᭤ࡆኚᙧࢆ㸪3D FEM ࣔࢹࣝ࡜ẚ㍑
ࡋ࡚㸪ࡑࡢጇᙜᛶࢆ᳨ドࡍࡿ㸬
ḟඖ᭷㝈せ⣲ࣔࢹࣝࡢ᭤ࡆ๛ᛶࡢ⿵ṇ
᳨ド࡟ඛ❧ࡕ㸪ཧ↷ࡍࡿ 3D FEM ࣔࢹࣝࡢ᭤ࡆ๛ᛶࢆࣖࣥࢢ⋡ࡢಟṇ࡟ࡼࡾ⿵ṇࡍࡿ㸬
ヲ⣽࡞ࣔࢹࣝ໬ࡀ࡞ࡉࢀࡓ㍈ࡢ 3D FEM ࣔࢹࣝࡣ㸪⠇Ⅼ⮬⏤ᗘࡀ 6 ࡛࠶ࡿࡓࡵ㸪㍈ࡢ᭤
ࡆࡸࡡࡌࡾ࡟ຍ࠼࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࡶ⾲⌧ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬ࡋ࠿ࡋ㸪せ⣲
ࡢࢱ࢖ࣉࡸ࣓ࢵࢩࣗࢧ࢖ࢬ➼࡟ࡼࡗ࡚ᅛ᭷್ゎᯒ⤖ᯝࡀኚ໬ࡍࡿ㸬౛࠼ࡤ㸪୰ᐇ෇ᙧ᩿㠃
㍈ࡢ 3D FEM ࣔࢹࣝࡢ⡆౽࡞ࣔࢹࣝ໬᪉ἲ࡜ࡋ࡚㸪㧗ࡉࢆ㍈ࡢ༙ᚄ㸪ᖜࢆ㍈ࡢ඲㛗࡜ࡋࡓ
㛗᪉ᙧࡢᖹ㠃ࢆ௵ពࡢᩘࡢ㛗᪉ᙧせ⣲࡟ศ๭ࡋ㸪ࡇࡢᖹ㠃ࢆ㍈ࡢ୰ᚰ㍈࿘ࡾ࡟ 360rᅇ㌿ࡉ
ࡏ࡚㸪භ㠃యࡢࢯࣜࢵࢻせ⣲ࢆᐃ⩏ࡍࡿ᪉ἲࡀ࠶ࡿ㸬ࡇࡢሙྜ㸪㍈୰ᚰ࡟㏆࠸せ⣲ࡀᴂᙧ
࡟㏆࠸ṇභ㠃య࠿ࡽ࠿ࡅ㞳ࢀࡓᙧ≧࡜࡞ࡿࡓࡵ㸪ᅛ᭷್ゎᯒࡢ⢭ᗘࡀపୗࡍࡿᜍࢀࡀ࠶ࡿ㸬
1D FEM ࣔࢹ࡛ࣝࡶせ⣲ศ๭ᩘ࡟ࡼࡗ࡚ᅛ᭷್ゎᯒ⤖ᯝࡀኚ໬ࡍࡿࡀ㸪╔┠ࡍࡿᅛ᭷࣮ࣔࢻ
࡟ᑐࡋ࡚༑ศ࡞せ⣲ᩘࢆ☜ಖࡍࢀࡤ≉࡟୍ᵝ᩿㠃㍈࡛ࡣ㠀ᖖ࡟㧗࠸⢭ᗘ࡛ᅛ᭷್ࢆồࡵࡿ
ࡇ࡜ࡀ࡛ࡁࡿ㸦௜㘓 A-2 ཧ↷㸧㸬ࡑࡇ࡛㸪ࡇࡇ࡛ࡣ୍ᵝ᩿㠃㍈ࡢ 1D FEM ࣔࢹࣝࡢ᭤ࡆ᣺ື
ࡢᅛ᭷᣺ືᩘࢆཧ↷ࡋ࡚㸪3D FEM ࣔࢹࣝࡢࣖࣥࢢ⋡ࢆಟṇࡍࡿ᪉ἲ࡛㧗⢭ᗘ࡞ 3D FEM ࣔ
ࢹࣝࢆసᡂࡍࡿ㸬௨ୗ࡟ࡑࡢᡭ㡰ࢆ♧ࡍ㸬
ࡲࡎ㸪඲㛗 350mm㸪እᚄ 20mm ࡢᅗ 3.10 ࡟♧ࡋࡓ୍ᵝ᩿㠃㍈ࡢ 3D FEM ࣔࢹࣝ࡟ࡘ࠸࡚㸪
⮬⏤ᨭᣢ≧ែࡢᅛ᭷್ゎᯒࢆ⾜ࡗ࡚ 4 ḟࡲ࡛ࡢ᭤ࡆ᣺ືࡢᅛ᭷᣺ືᩘࢆồࡵࡿ㸬せ⣲ศ๭
ࡣᚋ㏙ࡢᅗ 3.12 ࡢᑠᚄ㒊࡜ྠ୍࡜ࡍࡿ㸬ᮦᩱ≉ᛶࡣࣖࣥࢢ⋡ E=2.09×1011 N/m2㸪ᶓᙎᛶಀᩘ
G= 8.0335×1010 N/m2㸪࣏࢔ࢯࣥẚ Ȟ=0.3008㸪ᐦᗘ ȡ=7846 kg/m3 ࡜ࡋࡓ㸬
32
Fig. 3.10 3D
3 FEM moddel of shaft with
w uniform cross-section
c
࡟㸪ᅗ 3.10 ࡢ୍ᵝ᩿㠃㍈
ࡢ
㍈ࡢ 3D FEM
M ࣔࢹࣝ࡜ྠ
ྠ୍ᑍἲ㸪ྠ
ྠ୍ᮦᩱ≉ᛶ
ᛶࡢ㸪ᅗ 3.11 ࡢ 1D
ḟ࡟
FEM ࣔࢹࣝ࡟ࡘ
ࡘ࠸࡚⮬⏤ᨭᣢ≧ែࡢᅛ᭷
᭷್ゎᯒࢆ⾜
⾜ࡗ࡚ 4 ḟࡲ
ࡲ࡛ࡢ᭤ࡆ᣺
᣺ືࡢᅛ᭷᣺
᣺ືᩘ
ࢆồࡵ
ࡵࡿ㸬せ⣲ศ
ศ๭ࡣᚋ㏙ࡢ
ࡢᅗ 3.13 ࡜ྠ
ྠ୍࡜ࡍࡿ㸬
Fig. 3.11 1D FEM moddel of shaft with
w uniform cross-section
c
ࡉࡽ࡟㸪୍ᵝ࡞
࡞୰ᐇ෇ᙧ᩿
᩿㠃㍈ࡢ 1D F
FEM ࣔࢹࣝ
ࣝࡢ᭤ࡆ 1 ḟ㹼4 ḟࡢᅛ᭷
᭷᣺ືᩘࢆཧ
ཧ↷ࡋ
D FEM ࣔࢹࣝ
ࣝࡢࣖࣥࢢ⋡
⋡ࢆ⿵ṇࡍࡿ
ࡿ㸬ࣖࣥࢢ⋡
⋡ࡢ⿵ṇ࡟ࡣ
ࡣ㸪ᘧ㸦3.17㸧
㸧ࡢᅛ᭷ゅ᣺
᣺ືᩘ
࡚ 3D
Ȧn ࡜᭤ࡆ๛ᛶ EII ࡢ㛵ಀ࠿ࡽ
ࡽᚓࡽࢀࡿᘧ
ᘧ㸦3.18㸧ࢆ⏝
⏝࠸ࡿ㸬
2
Z n v EI 㸦3.17㸧
2
§ 1 ·
Ec ¨
¸ ˜E
©1 H ¹
㸦3.18㸧
E 㸸⿵ṇ๓
๓ࡢࣖࣥࢢ⋡
⋡ (N/m2)
E’ 㸸⿵ṇᚋ
ᚋࡢࣖࣥࢢ⋡
⋡ (N/m2)
H 㸸᭤ࡆ 1 ḟ㹼4 ḟࡢ
ࡢᅛ᭷ゅ᣺ື
ືᩘࡢᖹᆒㄗ
ㄗᕪ
3 ࡢ 1D FEM
F
ࣔࢹࣝ 㸦ཧ↷ࣔࢹࣝ
ࣝ㸧࡜ᅗ 3.10 ࡢ 3D FEM
M ࣔࢹࣝࡢ᭤
᭤ࡆ 1
⾲ 3.1 ࡣ㸪ᅗ 3.11
᣺ືᩘࡢẚ㍑
㍑ࢆ♧ࡋࡓࡶ
ࡶࡢ࡛࠶ࡿ㸬3D Model 0 ࡀࣖࣥࢢ⋡
⋡ࢆ⿵ṇࡍࡿ
ࡿ๓ࡢ
ḟ㹼4 ḟࡢᅛ᭷᣺
E=2.009×1011 N/m2 ࡜ࡋࡓࣔࢹ
ࢹࣝ࠶ࡾ㸪3D
D Model 1 ࡀࣖࣥࢢ⋡ࢆ
ࡀ
ᘧ㸦3.18㸧࡟
࡟ࡼࡗ࡚⿵ṇ
ṇࡋࡓ
33
E’=2.05175×1011 N/m2 ࡜ࡋࡓࣔࢹ࡛ࣝ࠶ࡿ㸬1D FEM ࣔࢹࣝ࡟ᑐࡍࡿ㸪ࣖࣥࢢ⋡ࢆ⿵ṇࡍࡿ
๓ࡢ 3D FEM ࣔࢹࣝࡢᅛ᭷᣺ືᩘࡢㄗᕪࡢ⤯ᑐ್ࡣ᭱኱࡛ 1.02%࡛࠶ࡿࡢ࡟ᑐࡋ࡚㸪ࣖࣥࢢ
⋡ࢆ⿵ṇࡋࡓᚋࡢ 3D FEM ࣔࢹࣝࡢᅛ᭷᣺ືᩘࡢㄗᕪࡢ⤯ᑐ್ࡣ᭱኱࡛ 0.09%࡜⣙ 1/11 ࡟
పῶࡋ࡚࠸ࡿ㸬
Table 3.1 Comparison of natural frequencies for bending modes of a shaft
with uniform cross-section between 1D model and 3D model
Unit: Hz
Mode
1D Model
3D Model 0 (Error)
3D Model 1 (Error)
1
743.69
750.04
(0.85%)
743.14
(0.07%)
2
2018.8
2037.6
(0.93%)
2018.84
(0.00%)
3
3873.7
3908.8
(0.90%)
3872.85
(0.02%)
4
6236.7
6300.5
(1.02%)
6242.59
(-0.09%)
᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘ
ࡇࡇ࡛ࡣ㸪ᅗ 3.12 ࡢ(a)࡜(b)࡟♧ࡍ 2 ✀㢮ࡢ㍈ࢆྲྀࡾୖࡆࡿ㸬
㍈ A ࡣ 2 ⟠ᡤ࡟ྠ୍ᚄࡢ኱ᚄ㒊ࢆᣢࡕ㸪㍈ B ࡣ 2 ⟠ᡤ࡟␗࡞ࡿᚄࡢ኱ᚄ㒊ࢆᣢࡘ㸬㍈ࡢ
ᑍἲࢆ௨ୗ࡟♧ࡍ㸬
࣭Shaft A㸹 ඲㛗 350mm㸪ᑠᚄ㒊እᚄ ȭ20mm㸪
኱ᚄ㒊እᚄ ȭ25.4mm㸦ȕ=0.7874㸧࡜ ȭ25.4mm㸦ȕ=0.7874㸧
኱ᚄ㒊ࡢ఩⨨㸸ᕥ➃࠿ࡽ 57.3mm㹼82.7mm㸦h/d0=1㸧㸪
162.3mm㹼187.7mm㸦h/d0=1㸧
࣭Shaft B㸹 ඲㛗 350mm㸪ᑠᚄ㒊እᚄ ȭ20mm㸪
኱ᚄ㒊እᚄ ȭ25.4mm㸦ȕ=0.7874㸧࡜ ȭ35.4mm㸦ȕ=0.5650㸧
኱ᚄ㒊ࡢ఩⨨㸸ᕥ➃࠿ࡽ 57.3mm㹼82.7mm㸦h/d0=1㸧㸪
162.3mm㹼187.7mm㸦h/d0=0.7175㸧
34
(a) Shaft A
(b) Shaft B
Fig. 3.12
3D FEM models of shhafts with step
pped cross-seections (Shaft
ft A & B)
ᅗ 3.12 ࡢ 3D FEM ࣔࢹࣝ
ࣝࡢ⮬⏤ᨭᣢ
ᣢ≧ែࡢᅛ᭷
᭷್ゎᯒ⤖ᯝ㸦᭤ࡆ 1 ḟ㹼
㹼4 ḟࡢᅛ᭷
᭷᣺ື
᪉ἲࢆ㐺⏝ࡋ
ࡋࡓሙྜࡢᅛ
ᅛ᭷᣺ືᩘࢆ
ࢆホ౯
ᩘ㸧ࢆཧ↷ࡋ࡚㸪ᅗ 3.13 ࡢ 1D FEM ࣔ ࢹࣝ࡟ᥦ᱌᪉
㸬㍈ A㸪㍈ B ඹ࡟㸪3D FEM ࣔࢹࣝ
ࣝࡢᮦᩱ≉ᛶ
ᛶࡣࣖࣥࢢ⋡
⋡ E’=2.051755×1011 N/m2㸦⿵ṇ
ࡋࡓ㸬
ᚋ㸧㸪࣏࢔ࢯࣥẚ
ẚ Ȟ=0.3008㸪ᐦᗘ ȡ=78466 kg/m3 ࡜ࡋ㸪1D FEM ࣔࢹࣝࡢᮦᩱ
ࣔ
ᩱ≉ᛶࡣࣖࣥ
ࣥࢢ⋡
E=2.009×1011 N/m2㸪࣏࢔ࢯࣥẚ
ẚ Ȟ=0.3008㸪ᐦᗘ ȡ=784
46 kg/m3 ࡜ࡋ
ࡋࡓ㸬࡞࠾㸪ᅗ 3.13 ࡢ 1D
D FEM
ࣔࢹࣝ
࡛ࣝࡣᕥ➃࠿
࠿ࡽ 5㸪10㸪17㸪22 ␒┠
┠ࡢせ⣲㛗ࢆ
ࢆᘧ㸦3.10㸧࡟ᚑࡗ࡚ 9d /32 ࡜ࡋ࡚࠸ࡿ㸬
(a) Shaft A
(b) Shaft B
Fig. 3.13
1D FEM models of shhafts with step
pped cross-seections (Shaft
ft A & B)
35
ḟ 4 ḟࡢᅛ᭷
᭷࣮ࣔࢻࢆ♧
♧ࡋࡓࡶࡢ࡛
࡛࠶ࡿ㸬
ᅗ 3.14 ࡣ㍈ A ࡢ᭤ࡆ 1 ḟ㹼
1sst mode
2nd m
mode
3rrd mode
4th m
mode
Fig. 3.14
Eigenmodees of Shaft A
⾲ 3.2 ࡣ㸪ᅗ 3.12
3
(a)ࡢ㍈┤
┤ᚄኚ໬㒊ࢆ
ࢆ᭷ࡍࡿ㍈ A ࡢ 3D FEM
M ࣔࢹࣝ㸦ཧ
ཧ↷ࣔࢹࣝ㸧࡜㸪
EM ࣔࢹࣝ࡟
࡟ࡘ࠸࡚᭤ࡆ 1 ḟ㹼4 ḟࡢ
ࡢᅛ᭷᣺ືᩘ
ᩘࡢẚ
ᅗ 3.113 (a)ࡢ㍈ A ࡢୗグࡢ 3 ࡘࡢ 1D FE
㍑ࢆ♧
♧ࡋࡓࡶࡢ࡛
࡛࠶ࡿ㸬
࣭11D Model 1㸸㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧࢆ⪃
⪃៖ࡋ࡞࠸ࣔࢹࣝ
࣭11D Model 2㸸ᑠᇼࡢ᪉ἲ
ἲ࡟ࡼࡗ࡚㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧࢆ⪃៖ ࡋࡓࣔࢹࣝ
࣭11D Model 3㸸ᥦ᱌᪉ἲ࡟
࡟ࡼࡗ࡚㍈┤
┤ᚄኚ໬㒊᩿
᩿㠃ࡢᙎᛶኚ
ኚᙧࢆ⪃៖ࡋ ࡓࣔࢹࣝ
Table 3.2
2 Comparisoon of natural frequencies of
o shaft A
Unit: Hz
M
Mode 3D Moodel
1D Model
M
1 (Erroor)
1D Model
M
2 (Errorr)
1D M
Model 3 (Errorr)
1
763.664
772.55
(1.177%)
772.5
55
(1.17%
%)
767.009
(0.45%
%)
2
20500.9
2064
4.1
(0.644%)
2064
4.1
(0.64%
%)
2055..0
(0.20%
%)
3
39644.3
4027
7.7
(1.600%)
4027
7.7
(1.60%
%)
3985..8
(0.54%
%)
4
63077.7
6382
2.3
(1.18 %)
6382
2.3
(1.18%
%)
6323..7
(0.25%
%)
36
㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋ࡞࠸ 1D Model 1 ࡛ࡣ᭱኱࡛ 1.60%ࡢㄗᕪࡀ࠶ࡿ㸬
ᅗ 3.13 (a)ࡢ㍈ A ࡢࣔࢹ࡛ࣝࡣ㸪ᘧ㸦1.1㸧ࡢ➼౯᭤ࡆ๛ᛶࢆ୚࠼ࡿ኱ᚄ㒊ࡢ㍈ᚄ de ࡀ኱ᚄ
㒊እᚄ d0 ࡼࡾ኱ࡁࡃ࡞ࡿࡓࡵ㸪ᑠᇼࡢ᪉ἲࡣ㐺⏝࡛ࡁ࡞࠸㸬ࡑࡢࡓࡵ㸪1D Model 2 ࡢᅛ᭷
᣺ືᩘࡣ 1D Model 1 ࡜ྠ୍ࡢ್࡜࡞ࡗ࡚࠸ࡿ㸬ᥦ᱌᪉ἲ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶ
ኚᙧࢆ⪃៖ࡋࡓ 1D Model 3 ࡢㄗᕪࡣ 0.54%௨ୗ࡛࠶ࡿ㸬
⾲ 3.3 ࡣ㸪ᅗ 3.12 (b)ࡢ㍈┤ᚄኚ໬㒊ࢆ᭷ࡍࡿ㍈ B ࡢ 3D FEM ࣔࢹࣝ㸦ཧ↷ࣔࢹࣝ㸧࡜㸪
ᅗ 3.13 (b)ࡢ㍈ B ࡢ๓㏙ࡢ 3 ࡘࡢ 1D FEM ࣔࢹࣝ࡟ࡘ࠸࡚᭤ࡆ 1 ḟ㹼4 ḟࡢᅛ᭷᣺ືᩘࡢẚ
㍑ࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬
Table 3.3 Comparison of natural frequencies of shaft B
Unit: Hz
Mode
3D Model
1D Model 1 (Error)
1D Model 2 (Error)
1D Model 3 (Error)
1
733.33
754.86
(2.94%)
739.64
(0.86%)
731.99
(-0.18%)
2
2034.8
2049.8
(0.74%)
2047.6
(0.63%)
2036.1
(0.06%)
3
3804.9
3924.8
(3.15%)
3841.9
(0.97%)
3812.8
(0.21%)
4
6161.8
6245.6
(1.36%)
6226.5
(1.05%)
6160.2
(-0.03%)
㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋ࡞࠸ 1D Model 1 ࡛ࡣ᭱኱࡛ 3.15%ࡢㄗᕪࡀ࠶ࡿ㸬
ᅗ 3.13 (b)ࡢ㍈ B ࡢࣔࢹࣝ࡟࠾࠸࡚㸪ᕥ➃࠿ࡽ 57.3mm㹼82.7mm ࡢ఩⨨࡟࠶ࡿ኱ᚄ㒊࡛ࡣᅗ
3.13 (a)ࡢ㍈ A ࡢࣔࢹࣝ࡜ྠᵝ࡟ᘧ㸦1.1㸧ࡢ➼౯᭤ࡆ๛ᛶࢆ୚࠼ࡿ኱ᚄ㒊ࡢ㍈ᚄ de ࡀ኱ᚄ㒊
እᚄ d0 ࡼࡾ኱ࡁࡃ࡞ࡿࡓࡵᑠᇼࡢ᪉ἲࡣ㐺⏝࡛ࡁ࡞࠸ࡀ㸪ᕥ➃࠿ࡽ 162.3mm㹼187.7mm ࡢ
఩⨨㸦୰ኸ㒊㸧࡟࠶ࡿ኱ᚄ㒊࡛ࡣ de ࡀ d0 ࡼࡾᑠࡉࡃ࡞ࡿࡓࡵ㸪ᑠᇼࡢ᪉ἲࢆ㐺⏝ࡍࡿࡇ࡜
ࡀ࡛ࡁࡿ㸦deq=31.41 mm㸧㸬ᑠᇼࡢ᪉ἲ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋࡓ 1D
Model 2 ࡛ࡣ㸪᭱኱࡛ 1.05%ࡢㄗᕪࡀ࠶ࡿ㸬ᥦ᱌᪉ἲ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ
ࢆ⪃៖ࡋࡓ 1D Model 3 ࡢㄗᕪࡣ 0.21%௨ୗ࡛࠶ࡿ㸬
࡞࠾㸪㍈ A ࡢ 1D Model 3 ࡼࡾ㸪㍈ B ࡢ 1D Model 3 ࡢ᪉ࡀᅛ᭷᣺ືᩘࡢㄗᕪࡀᑠࡉ࠸⌮
⏤ࡣ㸪㍈┤ᚄẚ ȕ ࡢᙳ㡪࡟ࡼࡿࡶࡢ࡜⪃࠼ࡽࢀࡿ㸬3.2 ⠇࡟♧ࡋࡓࡼ࠺࡟㸪㍈┤ᚄẚ ȕ ࡀ኱
ࡁ࠸࡯࡝ᘧ㸦1.4㸧࡟ࡼࡿ㏆ఝ⢭ᗘࡀపୗࡍࡿ㸬ࡑࡢࡓࡵ㸪ᥦ᱌᪉ἲࢆ㐺⏝ࡋࡓ 1D Model 3
࡛ࡣ㸪ȕ=0.7874 ࡢ㍈┤ᚄኚ໬㒊ࢆ 4 ⟠ᡤᣢࡘ㍈ A ࡼࡾ㸪ȕ=0.7874 ࡜ ȕ=0.5650 ࡢ㍈┤ᚄኚ໬
㒊ࢆ 2 ⟠ᡤࡎࡘᣢࡘ㍈ B ࡢ᪉ࡀ⢭ᗘࡢ㧗࠸ࣔࢹࣝ࡜࡞ࡗ࡚࠸ࡿ㸬
௨ୖ㸪ᅗ 3.13 ࡢ㍈┤ᚄኚ໬㒊ࢆ᭷ࡍࡿ 2 ࡘࡢ㍈㸪
㍈ A ࡜㍈ B ࡢ 1D FEM ࣔࢹࣝ࡟ࡘ࠸࡚㸪
ᅗ 3.12 ࡢ 3D FEM ࣔࢹࣝࢆཧ↷ࡋ࡚㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋ࡞࠸ሙྜ㸪ᑠ
ᇼࡢ᪉ἲ࡛⪃៖ࡋࡓሙྜ㸪ᥦ᱌᪉ἲ࡛⪃៖ࡋࡓሙྜࡢ 3 ࢣ࣮ࢫ࡛᭤ࡆ 1 ḟ㹼4 ḟࡢᅛ᭷᣺ື
ᩘࢆẚ㍑ࡋ㸪ᥦ᱌᪉ἲࡀጇᙜ࡛࠶ࡿࡇ࡜ࢆ☜ㄆࡋࡓ㸬
37
㍈┤ᚄኚ໬㒊
㒊ࢆ᭷ࡍࡿࡀ
ࡀ㸪㍈ A ࡛ࡣ
ࡣ㍈┤ᚄኚ໬
໬㒊ࡢ࡝ࡕࡽ
ࡽ࡟ࡘ
㍈ A㸪㍈ B ඹ࡟ 2 ⟠ᡤࡢ㍈
ἲࡀ㐺⏝࡛ࡁ
ࡁࡎ㸪㍈ B ࡛ࡣ㍈┤ᚄኚ
ኚ໬㒊ࡢ୍᪉
᪉࡟ࡘ࠸࡚ᑠ
ᑠᇼࡢ᪉ἲࡀ
ࡀ㐺⏝
࠸࡚ࡶᑠᇼࡢ᪉ἲ
࡞࠸ࡓࡵ㸪ྠ
ྠ୍᮲௳࡛ᑠ
ᑠᇼࡢ᪉ἲ࡜
࡜ẚ㍑ࡋࡓࡇ
ࡇ࡜࡟࡞ࡽ࡞
࡞࠸㸬ࡑࡇ࡛
࡛㸪ᑠᇼࡢ᪉
᪉ἲࢆ
࡛ࡁ࡞
㐺⏝࡛
࡛ࡁࡿ㍈┤ᚄ
ᚄኚ໬㒊ࢆ᭷
᭷ࡍࡿ㍈࡜ࡋ
ࡋ࡚ᅗ 3.15 ࡟♧ࡍ㍈
࡟
C ࡜㍈
࡜ D ࢆ⏝࠸
࠸࡚㸪ᑠᇼࡢ
ࡢ᪉ἲ
࡜ᥦ᱌
᱌᪉ἲ࡟ࡘ࠸
࠸࡚ᅛ᭷᣺ື
ືᩘࡢ⢭ᗘࡢ
ࡢẚ㍑ࢆ⾜ࡗ
ࡗࡓ㸬
(a) Shaft C
(b) Shaft D
Fig. 3.15
3D FEM models of shhafts with step
pped cross-seections (Shaft
ft C & D)
㍈ C ࡣ㸪ᅗ 3.112 ࡢ㍈ B ࡢᕥഃࡢ┤ᚄ
ࡢ
ᚄȭ25.4 ࡢ኱ᚄ
ᚄ㒊ࢆ↓ࡃࡋ࡚㸪኱ᚄ㒊
㒊ࢆ୰ኸࡢ┤
┤ᚄȭ
35.4 ࡢ
ࡢ㒊ศࡢࡳ࡜
࡜ࡋࡓ㍈࡛࠶
࠶ࡿ㸬㍈ D ࡣ
ࡣ㸪ᅗ 3.12 ࡢ㍈
ࡢ B ࡢᕥഃ
ഃࡢ┤ᚄȭ225.4 ࡢ኱ᚄ㒊
㒊ࡢ┤
ᚄࢆ㸪
㸪୰ኸࡢ኱ᚄ
ᚄ㒊࡜ྠ୍ࡢ
ࡢȭ35.4 ࡜ࡋ
ࡋࡓ㍈࡛࠶ࡿ
ࡿ㸬
⾲ 3.4 ࡣ㸪ᅗ 3.15
3
(a)ࡢ㍈┤
┤ᚄኚ໬㒊ࢆ
ࢆ᭷ࡍࡿ㍈ C ࡢ 3D FEM
M ࣔࢹࣝ㸦ཧ
ཧ↷ࣔࢹࣝ㸧࡜㸪
ࡢࣔࢹࣝ໬ࢆ
ࢆ⾜ࡗࡓ๓㏙ ࡢ 3 ࡘࡢ 1D
D FEM ࣔࢹࣝ࡟ࡘ࠸࡚㸪
㹼4 ḟ
ᅗ 3.113 ࡜ྠᵝࡢ
㸪᭤ࡆ 1 ḟ㹼
ࡢᅛ᭷
᭷᣺ືᩘࡢẚ
ẚ㍑ࢆ♧ࡋࡓ
ࡓࡶࡢ࡛࠶ ࡿ㸬㍈┤ᚄኚ
ኚ໬㒊᩿㠃ࡢ
ࡢᙎᛶኚᙧࢆ
ࢆ⪃៖ࡋ࡞࠸
࠸ 1D
Modeel 1 ࡛ࡣ᭱኱
኱࡛ 2.72%ࡢㄗᕪࡀ࠶ࡿ ࡀ㸪ᑠᇼࡢ᪉
᪉ἲࢆ㐺⏝ࡋ
ࡋࡓ 1D Modeel 2 ࡢㄗᕪࡢ
ࡢ⤯ᑐ
್ࡢ᭱
᭱኱್ࡣ 0.556%㸪ᥦ᱌᪉
᪉ἲࢆ㐺⏝ࡋ
ࡋࡓ 1D Mod
del 3 ࡢㄗᕪࡢ⤯ᑐ್ࡢ᭱
᭱኱್ࡣ 0.2
28%࡛
࠶ࡾ㸪
㸪࡝ࡕࡽࡶⰋ
Ⰻዲ࡞⢭ᗘࡀ
ࡀᚓࡽࢀ࡚࠸
࠸ࡿ㸬
38
Table 3.4 Comparison of natural frequencies of shaft C
Unit: Hz
Mode
3D Model
1D Model 1 (Error)
1D Model 2 (Error)
1D Model 3 (Error)
1
730.28
750.15
(2.72%)
729.84
(-0.06%)
728.62
(-0.23%)
2
2001.2
2005.6
(0.22%)
2003.1
(0.10%)
1999.2
(-0.10%)
3
3743.2
3832.4
(2.38%)
3722.2
(-0.56%)
3740.5
(-0.07%)
4
6080.9
6121.4
(0.67%)
6096.5
(0.26%)
6063.6
(-0.28%)
⾲ 3.5 ࡣ㸪ᅗ 3.15 (b)ࡢ㍈┤ᚄኚ໬㒊ࢆ᭷ࡍࡿ㍈ D ࡢ 3D FEM ࣔࢹࣝ㸦ཧ↷ࣔࢹࣝ㸧࡜㸪
ᅗ 3.13 ࡜ྠᵝࡢࣔࢹࣝ໬ࢆ⾜ࡗࡓ๓㏙ࡢ 3 ࡘࡢ 1D FEM ࣔࢹࣝ࡟ࡘ࠸࡚㸪᭤ࡆ 1 ḟ㹼4 ḟ
ࡢᅛ᭷᣺ືᩘࡢẚ㍑ࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋ࡞࠸ 1D
Model 1 ࡛ࡣ᭱኱࡛ 3.65%ࡢㄗᕪࡀ࠶ࡿࡀ㸪ᑠᇼࡢ᪉ἲࢆ㐺⏝ࡋࡓ 1D Model 2 ࡢㄗᕪࡢ⤯ᑐ
್ࡢ᭱኱್ࡣ 1.01%㸪ᥦ᱌᪉ἲࢆ㐺⏝ࡋࡓ 1D Model 3 ࡢㄗᕪࡢ⤯ᑐ್ࡢ᭱኱್ࡣ 0.38%࡛
࠶ࡾ㸪᫂ࡽ࠿࡟ᥦ᱌᪉ἲࡢ᪉ࡀ⢭ᗘࡀⰋዲ࡛࠶ࡿ㸬ᑠᇼࡢ᪉ἲ࡛ࡣ኱ᚄ㒊ࡢ㍈ᚄࢆῶࡌ࡚
㍈┤ᚄኚ໬㒊᩿㠃ࡢࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⾲⌧ࡋ࡚࠸ࡿࡢ࡟ᑐࡋ࡚㸪ᥦ᱌᪉ἲ࡛ࡣ
㍈ᑠᚄ㒊࡟タࡅࡓ㛗ࡉ 9d/32 ࡢせ⣲ࡢ᭤ࡆ๛ᛶࢆᙅࡵࡿࡇ࡜࡟ࡼࡗ࡚┤ᚄኚ໬㒊ࡢࡳ࠿ࡅ
ࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⾲⌧ࡋ࡚࠸ࡿ㸬ࡑࡢࡓࡵ㸪୧⪅ࡢᅛ᭷᣺ືᩘࡢண ⢭ᗘࡢᕪ␗ࡣ㸪
㍈┤ᚄኚ໬㒊᩿㠃ࡢࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⾲⌧ࡍࡿ㒊఩ࡀ␗࡞ࡿࡇ࡜࡟㉳ᅉࡍࡿ࡜
⪃࠼ࡽࡿ㸬
ᥦ᱌᪉ἲࢆ⏝࠸ࡿࡇ࡜࡟ࡼࡗ࡚㸪ᑠᇼࡢ᪉ἲࡼࡾ㧗⢭ᗘ࡟ᅛ᭷᣺ືᩘࢆண ࡛ࡁࡿࡇ࡜
ࡀ☜ㄆࡉࢀࡓࡀ㸪ᑠᇼࡢ᪉ἲࡶ㍈┤ᚄኚ໬㒊ࡢࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⪃៖ࡋ࡞࠸ሙ
ྜ࡜ẚ࡭ࡿ࡜༑ศ㧗⢭ᗘ࡜࠸࠼ࡿ㸬
Table 3.5 Comparison of natural frequencies of shaft D
Unit: Hz
Mode
3D Model
1D Model 1 (Error)
1D Model 2 (Error)
1D Model 3 (Error)
1
728.39
750.15
(2.99%)
729.84
(0.20%)
725.61
(-0.38%)
2
2007.2
2005.6
(-0.08%)
2003.1
(-0.20%)
2002.0
(-0.26%)
3
3697.5
3832.4
(3.65%)
3722.2
(0.67%)
3705.8
(0.22%)
4
6035.3
6121.4
(1.43%)
6096.5
(1.01%)
6035.8
(0.01%)
㟼Ⲵ㔜࡟ࡼࡿ᭤ࡆኚᙧࡢホ౯
ࡇࡇ࡛ࡣ㸪ᅗ 3.13 (b)ࡢ㍈ B ࡢ୰ኸࡢ኱ᚄ㒊࡜ྑഃࡢᑠᚄ㒊࠿ࡽ࡞ࡿ㍈㸪ࡍ࡞ࢃࡕᅗ 3.13
ࡢ㍈ B ࡢ⠇Ⅼ 18 ࠿ࡽ⠇Ⅼ 35 ࡲ࡛ࡢ㍈ࢆྲྀࡾୖࡆࡿ㸬኱ᚄ㒊እ࿘㠃ࢆᣊ᮰ࡋ࡚ᑠᚄ㒊ඛ➃
39
ࡆ࣮࣓ࣔࣥࢺࢆ୚࠼ࡓ࡜
࡜ࡁࡢࡓࢃࡳ
ࡳ᭤⥺ࢆ㸪ᅗ
ᅗ 3.16 ࡢ 3D
D FEM ࣔࢹࣝ
ࣝ࡜ᅗ 3.17 ࡢ 1D
࡟᭤ࡆ
FEM ࣔࢹࣝ࡟ࡘ࠸࡚ồࡵ࡚ẚ
ẚ㍑ࢆ⾜ࡗࡓ
ࡓ㸬ᅗ 3.17 ࡢ 1D FEM ࣔࢹ࡛ࣝࡣᕥ
ࣔ
ᕥ➃࠿ࡽ 5 ␒┠ࡢ
␒
せ⣲㛗
㛗ࢆ 9d/32 ࡜ࡋ࡚࠸ࡿ㸬
࡜
Fixed
M
Fiig. 3.16 3D FEM model of shaft withh stepped crosss-section forr bending defl
flection analy
ysis
Fixed
M
Fiig. 3.17 1D FEM model of shaft withh stepped crosss-section forr bending defl
flection analy
ysis
1D
D FEM ࣔࢹࣝ
࡛ࣝࡣ㸪㍈┤
┤ᚄኚ໬㒊᩿
᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋ࡞
࡞࠸ 1D Moddel 1㸪ᑠᇼࡢ
ࡢ᪉ἲ
࡟ࡼࡗ
ࡗ࡚㍈┤ᚄኚ
ኚ໬㒊᩿㠃ࡢ
ࡢᙎᛶኚᙧࢆ
ࢆ⪃៖ࡋࡓᥦ
ᥦ᱌᪉ἲࢆ㐺
㐺⏝ࡋࡓ 1D Model 2㸪ᥦ
ᥦ᱌᪉
ἲ࡟ࡼࡗ࡚㍈┤ᚄ
ᚄኚ໬㒊᩿㠃
㠃ࡢᙎᛶኚᙧ
ᙧࢆ⪃៖ࡋࡓ
ࡓ 1D Model 3 ࡢ 3 ࡘࡢࣔ
ࣔࢹࣝ࡟ࡘ࠸
࠸࡚ࡓ
ࡓ㸬ࡇࡇ࡛㸪ᑠᚄ㒊ඛ➃
➃࡟୚࠼ࡿ᭤
᭤ࡆ࣮࣓ࣔࣥ
ࣥࢺࡣ M=1000 N‚m ࡜ࡋࡓ
ࡓ㸬ࣖ
ࢃࡳ᭤⥺ࢆồࡵࡓ
⋡࡜࣏࢔ࢯࣥ
ࣥẚࡣ㸪3.4..2 ⠇࡜ྠࡌ್
್ࢆ⏝࠸ࡓ㸬
ࣥࢢ⋡
ᅗ 3.18 ࡣྛࣔࢹࣝࡢࡓࢃࡳ᭤⥺ࢆ♧ ࡋࡓࡶࡢ࡛࠶
࠶ࡿ㸬ྠᅗ࡟
࡟ࡣ z=20mm
m ࡲ࡛ࡢ㒊ศ
ศࡢᣑ
➃࡟⏕ࡌࡿ᭱
᭱኱ࡓࢃࡳࡢ
ࡢ್ࢆซ౛ࡢ
ࡢ㡰࡟♧ࡋࡓ
ࡓ㸬3D FEM
M ࣔࢹࣝ࡟ᑐ
ᑐࡍࡿ
኱ᅗ࡟
࡟ຍ࠼࡚ඛ➃
1D M
Model 1㸪1D Model
M
2㸪1D Model
M
3 ࡢ᭱
᭱኱ࡓࢃࡳࡢ⤯ᑐ್ࡢㄗᕪ
ᕪࡣࡑࢀࡒࢀ
ࢀ㸪-3.92%㸪-1.33%㸪
-0.10%
%࡛࠶ࡾ㸪ᥦ᱌᪉ἲ࡟ࡼ
ᥦ
ࡗ࡚㍈┤ᚄ ኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ
ࢆ⪃៖ࡍࡿࡇ
ࡇ࡜࡟ࡼࡗ࡚
࡚ࡓࢃ
ࡳ㔞ࡢ
ࡢㄗᕪࢆ኱ᖜ
ᖜ࡟పῶ࡛ࡁ
ࡁࡿࡇ࡜ࡀ☜
☜ㄆࡉࢀࡓ㸬
40
M=100 (Nm
m)
Max
ximum
defleection
-0.8
852 mm
-0.819 mm
-0.841 mm
-0.851 mm
Fig. 3.18
Deflectio
on of shaft wiith stepped crross-section due
d to bendingg moment
ࡼࡿ᭤ࡆ࣮ࣔ
࣮ࢻࡢᅛ᭷᣺
᣺ືᩘࡢホ౯
ᐇ㦂࡟ࡼ
ᅗ 3.12 (a)࡟♧
♧ࡉࢀࡿ㍈┤ᚄኚ໬㒊ࢆᣢ
ᣢࡘ㍈ A ࢆ㸪ྠ୍㒊ᮦ࠿
࠿ࡽษࡾฟࡋ
ࡋ࡚ษ๐ຍᕤ
ᕤ࡟ࡼ
ࡋࡓ㸬ࡇࡢ㍈ࡢ〇సᅗࢆ௜
௜㘓 A-3 ࡟♧
♧ࡍ㸬ᅗ 3.19 ࡣ⾪ᧁຍ᣺
᣺ヨ㦂ࡢ≧ἣ
ἣࢆ♧
ࡗ࡚ 16 ᮏ〇సࡋ
㸬㍈ཷྎ࡟⨨
⨨࠿ࢀࡓ㍈ࢆ
ࢆỈᖹ᪉ྥ࡟
࡟⾪ᧁຍ᣺ࡋ
ࡋࡓ࡜ࡁࡢຍ
ຍ᣺ຊ࡜㍈➃
➃ࡢ᣺
ࡋࡓࡶࡢ࡛࠶ࡿ㸬
㏿ᗘᛂ⟅࠿ࡽᚓࡽࢀࡿ࿘
࿘Ἴᩘᛂ⟅㛵
㛵ᩘࢆ㸪࿘Ἴ
Ἴᩘ㡿ᇦἲࡀ
ࡀ㐺⏝ࡉࢀࡓ
ࡓᐇ㦂࣮ࣔࢻ
ࢻゎᯒ
ືຍ㏿
ࢯࣇࢺ࢙࢘࢔㸦Balmes,
B
E, 19
996㸧ࢆ⏝࠸
࠸࡚ᴟ㸫␃ᩘ
ᩘࣔࢹࣝ㸦Alllemang, R. J.., 1994㸧࡟᭤
᭤⥺㐺
࡚୙ῶ⾶ᅛ᭷
᭷᣺ືᩘࢆồ
ồࡵࡓ㸬ᘧ㸦3.19㸧࡟ r ḟࡢ࣮ࣔࢻ࡟
ḟ
࡟㛵ࡍࡿ㸪」
」⣲ᅛ᭷್ Ȝr ࡜୙
ྜࡋ࡚
ῶ⾶ᅛ
ᅛ᭷ゅ᣺ືᩘ
ᩘ Ȧnr ࡢ㛵ಀ
ಀࢆ♧ࡍ㸬
Response Point
Impulse Ham
mmer
E
Excitation Poin
nt
Sponge
x
y
Accelerometer
z
Test Piecee
Peddestal
Fig. 3.119
Impact test configuraation of expeerimental mod
dal analysis oof a shaft
with sttepped corss--section
41
Or
9 r Z nr jZ dr 㸦3.19㸧
Ȧnr㸸r ḟࡢ
ࡢ୙ῶ⾶ᅛ᭷
᭷ゅ᣺ືᩘ ((rad/s)
ȗr㸸r ḟࡢ࣮ࣔࢻῶ⾶ẚ
ẚ
Ȧd r㸸r ḟࡢ
ࡢῶ⾶ᅛ᭷ゅ᣺
᣺ືᩘ (rad/s))㸪 Z dr
2
1 9 r Z nr
㍈ࡢ
ࡢᨭᣢ᪉ἲࡣ
ࡣ㸪㍈┤ゅỈ
Ỉᖹ᪉ྥࡢࡳ
ࡳࡢ㐠ືࢆチ
チᐜࡍࡿࡓࡵ
ࡵ࡟㍈ࡀỈᖹ
ᖹ᪉ྥ࡟ࢫ࣏
࣏ࣥࢪ
࡛ᙎᛶ
ᛶᨭᣢࡉࢀࡿᵓ㐀࡜ࡋ㸪ᦶ᧿ࡢᙳ㡪
㡪ࢆపῶࡍࡿ
ࡿࡓࡵ࡟㍈࡜
࡜㍈ཷྎࡢ᥋
᥋ゐ㠃࡟ࡣࢢ
ࢢࣜࢫ
ࢆሬᕸ
ᕸࡋࡓ㸦ᅗ 3.20 ཧ↷㸧㸬
Spring
S
(Sponge)
Shaaft
(Test piece)
p
Grease
G
Peddestal
Fig. 3.20
Support meth
hod of a shaft
ft
⾲ 3.6 ࡟ィ ᶵ
ᶵჾࡢ୍ぴࢆ
ࢆ♧ࡍ㸬ᐇ㦂
㦂࣮ࣔࢻゎᯒ
ᯒࢯࣇࢺ࢙࢘࢔࡜ࡋ࡚ࡣ 㸪MATLABpࡢࢧ
ࣃ࣮ࢸ࢕ Tooolbox ࡛࠶ࡿ
ࡿ SDTools ♫
♫〇ࡢ Structu
ural Dynamiccs Toolbox ࢆ
ࢆ౑⏝ࡋࡓ㸬
࣮ࢻࣃ
Table 3.6
Measuring instruments
Iteem
Manufacturer
M
Type
Impulse㻌 Hammer
PCB
T086C03
1
24130
Steel Tip
Accelerrometer
PCB
352A10
1
10335
0.7g
Dynamiic Signal
Anallyzer
Crystal
Instruments
I
Coco-80
1
36678
24bit
Ȉǻ ADC
42
Quantity
Q
Serrial Number
Notes
ᧁຍ᣺ヨ㦂࡟
࡟ࡼࡗ࡚ᚓࡽ ࢀࡓ࿘Ἴᩘᛂ
ᛂ⟅㛵ᩘ࡜࢝
࣮࢝ࣈࣇ࢕ࢵ
ࢵࢺ⤖ᯝࡢ౛
౛ࢆ♧
ᅗ 3.21 ࡟⾪ᧁ
࡚㸪㟷Ⰽࡢᐇ
ᐇ⥺ࡀᐇ ࡉ
ࡉࢀࡓ࿘Ἴᩘ
ᩘᛂ⟅㛵ᩘ࡛
࡛࠶ࡾ㸪⥳Ⰽ
Ⰽࡢᐇ⥺ࡀ࢝
࣮࢝ࣈ
ࡍ㸬ྠᅗ࡟࠾࠸࡚
࡚ᚓࡽࢀࡓ࿘
࿘Ἴᩘᛂ⟅㛵
㛵ᩘ࡛࠶ࡿ㸬
ࣇ࢕ࢵࢺ࡟ࡼࡗ࡚
㸫 Test
㸫 Fit
Fig. 3.21
An example
e
for ccurve fitting of
o frequency response func
nction
⾲ 3.7 ࡟㸪ᅗ 3.12 (a)࡟♧ࡉ
ࡉࢀࡿ㍈┤ᚄ
ᚄኚ໬㒊ࢆᣢ
ᣢࡘ㍈ A ࡢ㸪1 ḟ㹼4 ḟࡢ
ࡢ᭤ࡆ࣮ࣔࢻ
ࢻࡢᅛ
ືᩘࡢᐇ㦂⤖
⤖ᯝࢆ♧ࡍ㸬ྠ⾲࡟ࡣ㸪 ྛ࣮ࣔࢻࡢ
ࡢᅛ᭷᣺ືᩘ
ᩘࡢᖹᆒ್㸪 ศᩓ ı2㸪3ı ࡢ್
᭷᣺ື
࡟ࡘ࠸
࠸࡚ࡶ♧ࡋ㸪3ı ࡢ್࡟
࡟ࡘ࠸࡚ࡣᅛ ᭷᣺ືᩘࡢᖹ
ᖹᆒ್࡟ᑐࡍ
ࡍࡿⓒศ⋡ࡢ
ࡢ್ࡶేグࡋ
ࡋࡓ㸬
ᐇ㦂࡛
࡛ᚓࡽࢀࡓᅛ
ᅛ᭷᣺ືᩘࡣ
ࡣ㸪16 ᮏࡢ ㍈ࡢ〇సㄗᕪ
ᕪ㸪࿘Ἴᩘᛂ
ᛂ⟅㛵ᩘࡢィ
ィ ㄗᕪ㸪࢝
࣮࢝ࣈ
ࣇ࢕ ࢵࢺࡢㄗᕪࢆྵࢇࡔࡶࡢ࡜⪃࠼ࡿ ࡇ࡜ࡀ࡛ࡁࡿࡀ㸪16 ᮏࡢ㍈ࡢᅛ᭷
ᮏ
᭷᣺ືᩘࡢᖹ
ᖹᆒ್
764.556 Hz㸪2053.99 Hz㸪3966.7
7 Hz㸪6313.77 Hz ࡣ㸪⾲ 3.2
3 ࡢ 3D Mo
odel ࡢᅛ᭷᣺
᣺ືᩘ 763.64
4Hz㸪
2050..9Hz㸪3964.33Hz㸪6307.7
7Hz ࡟ᑐࡋ࡚
࡚㸪㧗ࠎ0.12
2%㸪0.15%㸪0.06%㸪0.110%ࡢᕪ␗ࡀ
ࡀ࠶ࡿ
࡟㐣ࡂ
ࡂ࡞࠸㸬1 ḟ㹼
ḟ 4 ḟࡢ᭤
᭤ࡆ࣮ࣔࢻࡢ ᅛ᭷᣺ືᩘࡢ 3ı ࡣ 0.5%
%ᮍ‶࡛࠶ࡿ
ࡿ㸬
43
Table 3.7
Identified natural frequencies of a shaft with stepped corss-section
under free-free support condition
Unit㸸Hz
1st mode
2nd mode 3rd mode
4th mode
Rotor 1
764.23
2052.2
3963.9
6309.0
Rotor 2
762.59
2050.1
3960.6
6304.4
Rotor 3
764.37
2054.0
3966.0
6313.3
Rotor 4
765.77
2055.0
3968.1
6313.5
Rotor 5
763.97
2052.2
3964.5
6307.5
Rotor 6
764.52
2054.5
3968.0
6316.8
Rotor 7
765.19
2054.9
3970.1
6315.6
Rotor 8
765.64
2055.0
3967.5
6314.0
Rotor 9
762.50
2048.7
3958.8
6302.0
Rotor10
766.25
2058.2
3973.6
6324.7
Rotor11
765.56
2055.3
3970.3
6318.1
Rotor12
764.20
2055.0
3970.0
6322.2
Rotor13
764.74
2054.5
3967.1
6315.0
Rotor14
765.26
2058.1
3973.8
6324.7
Rotor15
763.59
2049.5
3957.9
6300.9
Rotor16
764.59
2054.7
3967.3
.06316.8
Mean
764.56
2053.9
3966.7
6313.7
1.140
7.380
21.90
53.74
3.203
8.150
14.04
21.99
(0.42%)
(0.40%)
(0.35%)
(0.35%)
2
ı
3ı
(Percentage
against
mean value)
⾲ 3.8 ࡣ㸪㍈ A ࡢ 1 ḟ㹼4 ḟࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࢆᐇ㦂⤖ᯝ㸦ᖹᆒ್㸧࡜ 1D FEM
ࣔࢹࣝ࡟ࡘ࠸࡚♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬ྠ⾲ࡣ㸪⾲ 3.2 ࡢཧ↷್ࢆ 3D FEM ࣔࢹࣝࡢ್࠿ࡽ⾲
3.7 ࡢᐇ㦂⤖ᯝࡢᖹᆒ್࡟ኚ᭦ࡋࡓࡶࡢ࡛࠶ࡿ㸬ࡇࡇ࡛㸪㍈ A ࡟ᑠᇼࡢ᪉ἲࡣ㐺⏝࡛ࡁ࡞࠸
ࡓࡵ 1D Model 2 ࡣ┬࠸ࡓ㸬ᐇ㦂⤖ᯝ㸦ᖹᆒ್㸧࡟ᑐࡍࡿ㸪ᥦ᱌ࡍࡿ㍈┤ᚄኚ໬㒊ࡢࣔࢹࣝ
ᐇ㦂⤖ᯝ࠿ࡽࡶ㸪
໬᪉ἲࢆ㐺⏝ࡋࡓ 1D Model 3 ࡢᅛ᭷᣺ືᩘࡢㄗᕪࡣ᭱኱࡛ 0.48%࡛࠶ࡾ㸪
ᥦ᱌ࡍࡿ㍈┤ᚄኚ໬㒊ࡢࣔࢹࣝ໬᪉ἲࡀጇᙜ࡛࠶ࡿࡇ࡜ࡀ☜ㄆࡉࢀࡓ㸬
44
Tabble 3.8 1st to
t 4th naturall frequencies of bending modes
m
for shaaft A
(Tesst and 1D FEM㸧
Unit 㸸Hz
Testt
Moode
Model 1 (Error)
1D Model
M
3 (Errorr)
1D M
(Meaan)
1
764.5
56
772. 55
(1.05%)
767.0
09
(0.33%
%)
2
2053.9
20644.1
(0.50%)
2055.0
(0.05%
%)
3
3966.7
40277.7
(1.54%)
3985.8
(0.48%
%)
4
6313.7
63822.3
(1.09%)
6323.7
(0.16%
%)
ᐇᶵ┦ᙜ
ᙜࡢゎᯒࣔࢹ
ࢹࣝ࡟ࡼࡿ᭤ ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡢ
ࡢホ౯
ᮏ⠇
⠇࡛ࡣ㸪ᅗ 3.22 ࡟♧ࡍ
ࡍࢱ࣮ࣅࣥ㸫Ⓨ
Ⓨ㟁ᶵ࣮ࣟࢱࡢᐇᶵ┦ᙜ
ᙜࣔࢹࣝࢆ⏝
⏝࠸࡚㸪㍈┤
┤ᚄኚ
໬㒊᩿
᩿㠃ࡢᙎᛶኚ
ኚᙧࢆᥦ᱌᪉
᪉ἲ࡟ࡼࡗ࡚
࡚⪃៖ࡋࡓሙ
ሙྜ࡜⪃៖ࡋ
ࡋ࡞࠸ሙྜ࡟
࡟ࡘ࠸࡚㸪᭤
᭤ࡆࣔ
࣮ࢻࡢ
ࡢᅛ᭷್ゎᯒ
ᯒ⤖ᯝࢆẚ㍑
㍑ࡍࡿ㸬
Tuurbine
Coupling
Generator
Exciter
E
Fig. 3.22
Ben
nding vibratioon analysis model
m
of turbin
ne – generatoor rotor
ᅗ 3.22 ࡢࣔࢹ
ࢹࣝࡣ㸪ࢱ࣮ࣅ࣮ࣥࣟࢱ ࡜Ⓨ㟁ᶵ࣮ࣟ
࣮ࢱࡀ࢝ࢵࣉ
ࣉࣜࣥࢢ࡛๛
๛⤖ྜࡉࢀࡓ
ࡓ1㍈
ࢱࡀ⠇Ⅼ␒ྕ
ྕ 6㸪45㸪67
7㸪99㸪107 ࡢ఩⨨࡛ࣆࣥᨭᣢࡉࢀࡓ
ࡓᵓ㐀ࢆᣢࡗ
ࡗ࡚࠸ࡿ㸬ࢱ
ࢱ࣮ࣅ
࣮ࣟࢱ
ࣥ㸫Ⓨ
Ⓨ㟁ᶵ࣮ࣟࢱ
ࢱࡣ඲㛗 29,,614mm㸪⠇
⠇Ⅼᩘ 108㸪㍈
㍈ࡢせ⣲ᩘ 107 ࡢᅇ㌿㍈
㍈࡜㸪ࣈ࣮ࣞ
࣮ࢻ➼
ࡢࢹ࢕ࢫࢡせ⣲࡛
࡛ᵓᡂࡉࢀ࡚
࡚࠸ࡿ㸬㯤Ⰽ
Ⰽ࡛♧ࡋࡓ㸪せ⣲␒ྕ 4㸪7㸪9㸪21㸪28㸪33㸪39㸪41㸪
43㸪550㸪53㸪59㸪62㸪68㸪74
7 㸪91㸪97㸪 102㸪105 ࡢせ⣲ࡣ㸪㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧ
࡟ࡼࡿࡳ࠿ࡅࡢ㍈
㍈ࡢ᭤ࡆ๛ᛶ
ᛶపୗࢆ⪃៖
៖ࡍ࡭ࡁせ⣲
⣲࡛࠶ࡿࡓࡵ
ࡵ㸪せ⣲㛗ࢆ 9d/32 ࡜ࡋ࡚
࡚࠸ࡿ㸬
࡛㸪せ⣲␒ྕ
ྕ 33 ࡟ࡘ࠸
࠸࡚ࡣ㸪せ⣲
⣲␒ྕ 33 ࡜ 34
3 ࡢせ⣲㛗ࢆ㊊ࡋྜࢃࡏ
ࡏࡓ್ࡀ 9d/32 ࡜
ࡇࡇ࡛
࡞ࡿࡼ࠺࡟タᐃࡋ
ࡋ࡚࠸ࡿ㸬௜
௜㘓 A-4 ࡟ ࢱ࣮ࣅࣥ㸫Ⓨ
Ⓨ㟁ᶵ࣮ࣟࢱ
ࢱࡢ᭤ࡆ᣺ື
ືゎᯒࣔࢹࣝ
ࣝࡢㅖ
♧ࡍ㸬
ඖࢆ♧
⾲ 3.9 ࡣ㸪㯤Ⰽ
Ⰽ࡛♧ࡋࡓせ
せ⣲ࡢ᭤ࡆ๛
๛ᛶࢆ E Ieq㸦ᘧ㸦
㸦
3.14㸧ཧ
ཧ↷㸧࡜ࡍࡿ
ࡿᥦ᱌᪉ἲ࡜
࡜㍈┤
໬㒊ࡢࡳ࠿ࡅ
ࡅࡢ㍈ࡢ᭤ࡆ
ࡆ๛ᛶపୗࢆ
ࢆ⪃៖ࡋ࡞࠸
࠸㏻ᖖࡢ᪉ἲ
ἲ࡟ࡘ࠸࡚㸪᭤
᭤ࡆ 1 ḟ㹼5 ḟࡢ
ᚄኚ໬
ᅛ᭷᣺
᣺ືᩘࡢィ⟬
⟬⤖ᯝࢆ♧ࡋࡓࡶࡢ࡛࠶
࠶ࡿ㸬ྠ⾲࡟
࡟ࡣ㸪ᥦ᱌᪉
᪉ἲ࡜㏻ᖖࡢ
ࡢ᪉ἲࡢᕪ␗
␗࡟ࡘ
45
࠸࡚ࡶ♧ࡋࡓ㸬ᥦ᱌ἲ࡟ࡼࡿ᭤ࡆ 3 ḟࡢᅛ᭷᣺ືᩘࡣ㸪㏻ᖖࡢ᪉ἲࡼࡾ⣙ 5%㧗࠸್࡛࠶ࡿ㸬
ཧ⪃࡟㸪᭤ࡆ 1 ḟ㹼5 ḟࡢᅛ᭷࣮ࣔࢻࢆᅗ 3.23 ࡟♧ࡍ㸬
Table 3.9
Natural frequencies of bending modes
for turbine – generator rotor
Unit: Hz
Mode Proposed method
Ordinary method
1
17.026
17.250
(1.32%)
2
17.509
17.708
(1.13%)
3
42.803
44.759
(4.57%)
4
62.921
64.389
(2.33%)
5
65.090
67.220
(3.27%)
2
1st
2nd
3rd
4th
5th
1.5
Deflection
1
0.5
0
-0.5
-1
-1.5
-2
0
5
10
15
20
25
30
Position of nodes (m)
Fig. 3.23
Eigenmodes of bending vibration for turbine – generator rotor
⤖ゝ
1 ḟඖᱱࡢ᭷㝈せ⣲ࣔࢹࣝ࡬ࡢ㐺⏝ࡀᐜ᫆࡞㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅ
ࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝ໬᪉ἲ࡟ࡘ࠸᳨࡚ウࡋ㸪௨ୗࡢ⤖ㄽࢆᚓࡓ㸬
(1) ኱ᚄ㒊ࡀཌᯈ࡜ࡳ࡞ࡏࡿ㸪ᙎᛶኚᙧࢆ⪃៖ࡍ࡭ࡁ኱ᚄ㒊ࡢᖜᚄẚ h/d0 ࡢ⠊ᅖ㸪ࡍ࡞ࢃ
ࡕᥦ᱌᪉ἲࡢ㐺⏝⠊ᅖࡣ㸪㍈┤ᚄẚ ȕ ࡟ࡼࡽࡎ࠾ࡼࡑ h/d0 • 0.2 ࡛࠶ࡿ㸬
(2) ᑠᚄ㒊࡟㛗ࡉ 9d/32 ࡢ㍈せ⣲ࢆタࡅ㸪ࡑࡢ㒊ศࡢ᭤ࡆ๛ᛶࢆᘧ㸦3.20㸧࡛୚࠼ࡽࢀࡿ➼
౯࡞᭤ࡆ๛ᛶ Ieq ࡜ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈
46
ࡢ᭤ࡆ๛ᛶపୗࢆ⾲⌧ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
2
4
2
2
3
I 1 I 2 I1 I 1 7 I1 I 2 6I 1 I 2 2I 2
I eq
4
2I 1 I 2 㸦3.20㸧
I1㸸኱ᚄ㍈ࡢ᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ (m4)
I2㸸ᑠᚄ㍈ࡢ᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ(m4)
(3) ኱ᚄ㒊ࡀཌᯈ࡜ࡳ࡞ࡏࡿ㸪ᙎᛶኚᙧࢆ⪃៖ࡍ࡭ࡁ⠊ᅖ࡛ࡣ㸪኱ᚄ㍈┤ᚄ d0 ࡀᑠᚄ㍈┤
ᚄ d ࡟ᑐࡋ࡚࠸ࡃࡽ኱ࡁࡃ࡞ࡗ࡚ࡶ㸪➼౯࡞᭤ࡆ๛ᛶ EIeq ࡣᑠᚄ㍈ࡢ᭤ࡆ๛ᛶ EI2 ࡢ
0.4979 ಸࢆୗᅇࡽ࡞࠸㸬
(4) ᥦ᱌᪉ἲ࡟ࡼࡿᅛ᭷᣺ືᩘࡢゎᯒ⤖ᯝࡣ㸪ᚑ᮶⏝࠸ࡽࢀ࡚ࡁࡓᑠᇼࡢ᪉ἲࡼࡾ㧗⢭ᗘ
࡛࠶ࡿ㸬
(5) ᥦ᱌᪉ἲࢆ㐺⏝ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪ືⓗ⌧㇟࡛࠶ࡿᅛ᭷᣺ື≉ᛶࡔࡅ࡛࡞ࡃ㸪㟼ⓗ࡞
᭤ࡆኚᙧ࡟ࡘ࠸࡚ࡶ㧗⢭ᗘ࡞ゎᯒ⤖ᯝࢆᚓࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
(6) ᐇᶵ┦ᙜࡢࢱ࣮ࣅࣥ㸫Ⓨ㟁ᶵ࣮ࣟࢱࡀ 5 ⟠ᡤ࡛ࣆࣥᨭᣢࡉࢀࡓࣔࢹࣝ࡟࠾ࡅࡿ㸪పḟ
࣮ࣔࢻࡢᅛ᭷᣺ືᩘ࡟㛵ࡍࡿ㏻ᖖࡢ᪉ἲ࡜ᥦ᱌᪉ἲࡢᕪ␗ࡣ㸪᭱኱࡛⣙ 5%࡛࠶ࡗࡓ㸬
ཧ⪃ᩥ⊩
Allemang, R. J., Vibrations: experimental modal analysis, UC-SDRL-CN-20-263-662 (1994).
Balmes, E., Frequency domain identification of structural dynamics using the pole/residue
parametrization, International Modal Analysis Conference, (1996).
୰ἑ ୍, 㛗ᒇ ஧㑻, ຍ⸨ ༤, ᭱᪂ᶵᲔᕤᏛㅮᗙ ᮦᩱຊᏛ, ⏘ᴗᅗ᭩ (1973), p. 111.
㧗⏣ ᫀ஧, ⯚➉ Ⰻ㞝, ✄ᇉ ჆அ, ㍈┤ᚄኚ໬㒊ࢆ᭷ࡍࡿ࣮ࣟࢱࡢ᭤ࡆ᣺ື࣮ࣔࢻ࡟㛵ࡍ
ࡿ᣺ືィ⟬ࡢ㧗⢭ᗘ⡆᫆ࣔࢹࣝࡢᥦ᱌, ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟 C ⦅, Vol.75, No.752 (2009),
pp.824-829.
47
➨❶
❶ ᔐࡵྜ࠸
࠸㒊ࡢ᭤ࡆ๛
๛ᛶࡢ᳨ウ
⥴ゝ
⯡࡟ࢹ࢕ࢫࢡ
ࢡせ⣲࡜ࡋ࡚
࡚ᢅࢃࢀࡿࣈ
ࣈ࣮ࣞࢻࡸ࢖
࢖ࣥ࣌ࣛ࡞࡝
࡝ࡀ㍈࡟ᔐࡵ
ࡵྜ࠸࡟ࡼࡗ
ࡗ࡚ྲྀ
୍⯡
ࡾ௜ࡅ
ࡅࡽࢀࡓ࣮ࣟ
࣮ࢱࡢ㍈᣺ື
ືゎᯒࢆ⾜࠺
࠺ሙྜ㸪ᔐࡵ
ࡵྜ࠸㒊ࡢ᭤
᭤ࡆ๛ᛶࡢホ
ホ౯᪉ἲ࡟ࡼࡗ࡚㸪
᭤ࡆࣔ
࣮ࣔࢻࡢᅛ᭷
᭷᣺ືᩘ࡜ᅛ
ᅛ᭷࣮ࣔࢻࡀ
ࡀኚࢃࡗ࡚ࡋ
ࡋࡲ࠺㸬౛࠼
࠼ࡤ㸪ᔐࡵྜ
ྜࢃࡉࢀࡿࢹ
ࢹ࢕ࢫ
ࢡせ⣲
⣲ࡀ㍈ࡢ᭤ࡆ
ࡆ๛ᛶ࡟඲ࡃ
ࡃᐤ୚ࡋ࡞࠸
࠸࡜⪃࠼ࡿሙ
ሙྜࡣ㸪➼౯
౯࡞᭤ࡆ๛ᛶ
ᛶࢆ୚࠼ࡿ㍈
㍈ᚄࡣ
㍈┤ᚄ
ᚄ࡟୍⮴ࡍࡿ㸬୍᪉㸪ᔐ
ᔐࡵྜࢃࡉࢀ
ࢀࡿࢹ࢕ࢫࢡ
ࢡせ⣲ࡀ㍈ࡢ
ࡢ᭤ࡆ๛ᛶ࡟
࡟ᐤ୚ࡍࡿ࡜
࡜⪃࠼
ࡿሙྜ
ྜࡣ㸪➼౯࡞
࡞᭤ࡆ๛ᛶࢆ
ࢆ୚࠼ࡿ㍈ᚄ
ᚄࡣ㍈┤ᚄࡼ
ࡼࡾ኱ࡁࡃ࡞
࡞ࡿ㸬ࡋ࠿ࡋ
ࡋ㸪㍈┤ᚄ࡟
࡟ᑐࡋ
࡚➼౯
౯࡞᭤ࡆ๛ᛶ
ᛶࢆ୚࠼ࡿ㍈
㍈ᚄࢆ࡝ࡢ⛬
⛬ᗘ኱ࡁࡃࡍ
ࡍࢀࡤⰋ࠸࠿
࠿࡟ࡘ࠸᳨࡚
᳨ドࡋࡓ౛ࡣ
ࡣⴭ⪅
ࡢ▱ࡿ㝈ࡾぢᙜࡓ
ࡓࡽ࡞࠸㸬ࡑ
ࡑࡇ࡛㸪ᮏ❶
❶࡛ࡣ㸪ᔐࡵ
ࡵྜ࠸࡟ࡼࡗ
ࡗ࡚㍈ࡢ᭤ࡆ
ࡆ๛ᛶࡀ࡝ࡢ
ࡢ⛬ᗘ
ࡍࡿ࠿࡟ࡘ࠸
࠸࡚㸪ゎᯒ࡜
࡜ᐇ㦂࡛ホ౯
౯ࡋࡓ⤖ᯝࢆ
ࢆ♧ࡍ㸬
ቑ኱ࡍ
㒊ࡢ᭤ࡆ๛ᛶ
ᛶࡢホ౯
ᔐࡵྜ࠸㒊
ᯒ࡟ࡼࡿᔐࡵ
ࡵྜ࠸ᙉࡉ࡜ ㍈ࡢ᭤ࡆ๛ᛶࡢ㛵ಀࡢ᳨
᳨ウ
᥋ゐゎᯒ
㍈࡟
࡟ࢫ࣮ࣜࣈࡀ
ࡀᔐࡵྜࢃࡉ
ࡉࢀࡓᵓ㐀≀
≀࡟ࡘ࠸࡚㸪ᔐࡵྜ࠸ᙉ
ᙉࡉ࡜㍈ࡢ᭤
᭤ࡆ๛ᛶࡢ㛵ಀࢆ㸪
ᔐࡵྜ
ྜ࠸㒊ࡢᦶ᧿
᧿ࢆ⪃៖ࡋࡓ
ࡓ㠀⥺ᙧ㟼ゎ
ゎᯒ࡟ࡼࡗ࡚
࡚ィ⟬ࡋࡓ㸬
㛗ࡉ 350mm㸪┤ᚄ ȭ25.4m
mm ࡢ୰ᐇ෇
෇ᙧ᩿㠃ࡢ㍈
㍈ࡢ୰ኸ㒊࡟
࡟㸪㍈᪉ྥ㛗
㛗ࡉ 25.4mm㸪ཌࡉ
5.0mm
m ࡢࢫ࣮ࣜ
࣮ࣈࡀᔐࡵྜ࠸࡟ࡼࡗ࡚ྲྀ
ྲྀࡾ௜ࡅࡽࢀ
ࢀ࡚࠸ࡿᵓ㐀
㐀≀ࢆ᝿ᐃࡋ
ࡋ㸪Ⲵ㔜࡜ᙧ
ᙧ≧ࡢ
ᑐ⛠ᛶ
ᛶࢆ⪃៖ࡋ࡚
࡚㸪ᅗ 4.1 ࡢ 1/2 ࣔࢹࣝ
ࣝࢆゎᯒࣔࢹࣝ
ࣝ࡜ࡋࡓ㸬
ȭ25.4mm
5mm
175mm
12.7mm
M
y
x
Fig. 4.1
z
Analysis model
m
for inveestigating the influence of fit on bendinng stiffness
ゎᯒ
ᯒࢶ࣮ࣝ࡜ࡋ
ࡋ࡚ࡣ ABAQUS 6.12-2 ࢆ౑⏝ࡋ㸪㍈
㍈࡜ࢫ࣮ࣜࣈ
ࣈࡢᮦᩱ≉ᛶ
ᛶ್ࡣඹ࡟ࣖ
ࣖࣥࢢ
⋡ E=
=2.05×1011 N/m
N 2㸪࣏࢔ࢯ
ࢯࣥẚ Ȟ=0.30008 ࡜ࡋࡓ㸬
Ⲵ㔜
㔜᮲௳ࡣ㸪ᕥ
ᕥ➃㍈୰ᚰ࡟
࡟ x ㍈࿘ࡾࡢ
ࡢ࣮࣓ࣔࣥࢺ
ࢺ M ࢆ୚࠼ࡿ
ࡿࡶࡢ࡜ࡋ㸪 ᣊ᮰᮲௳ࡣ
ࡣ㸪ᅗ
48
4.1 ࡢ
ࡢᕥ➃㍈୰ᚰ
ᚰࡢ⠇Ⅼࡢ୪
୪㐍⮬⏤ᗘࢆ
ࢆᣊ᮰㸦ࣆࣥᨭᣢ㸧㸪ྑ➃
➃ࡢᑐ⛠㠃ࡢ
ࡢ㍈᪉ྥ㸦z ᪉ྥ㸧
ኚ఩ࢆᣊ᮰ࡋࡓ㸬ࡇࡢࣔࢹࣝ
ࣝࡣ㸪୧➃ࡀ
ࡀࣆࣥᨭᣢࡉ
ࡉࢀࡓ㛗ࡉ 350mm ࡢ㍈ࡢ
ࡢ 1/2 ࣔࢹࣝ
࡛ࣝ࠶
ࡿ㸬
ࡇࡢ
ࡢࣔࢹࣝ࡟࠾
࠾ࡅࡿᕥ➃࡜
࡜ྑ➃ࡢ y ᪉
᪉ྥࡢ┦ᑐኚ
ኚ఩ࡣ㸪⮬⏤
⏤➃࡟࣮࣓ࣔ ࣥࢺ M ࢆཷ
ཷࡅࡿ
∦ᣢࡕ
ࡕᱱࡢࡓࢃࡳ
ࡳ࡜୍⮴ࡍࡿ
ࡿࡓࡵ㸪ᔐࡵ
ࡵྜ࠸㒊ࡢ➼
➼౯᭤ࡆ๛ᛶ
ᛶ EIeq ࡣ㸪ᅗ 4.2 ࡟♧ࡉࢀ
ࢀࡿࢫ
࣮ࣜࣈ
ࣈ➃㒊ࡢ┦ᑐ
ᑐࡓࢃࡳ į0-į1 ࢆ∦ᣢࡕ ᱱࡢࡓࢃࡳࡢᘧ࡟㐺⏝ࡋ࡚ᚓࡽࢀࡿ
ࡿ㸪ᘧ(4.1)࡟ࡼࡗ
࡟
࡚ồࡵ
ࡵࡿࡇ࡜ࡀ࡛
࡛ࡁࡿ㸬
Ml 2
2G 0 G1 EI eq
(4.1)
y
x
z
į1
į0
l
Fig
g. 4.2
Defflection of shaaft with a sleeeve
ࡲࡓ
ࡓ㸪ᔐࡵྜ࠸
࠸㒊ࡢ➼౯࡞
࡞᭤ࡆ๛ᛶࢆ
ࢆ୚࠼ࡿ㍈ᚄ
ᚄ deq ࡣ㸪ᘧ(4.1)
(
ࡢ➼౯᩿
᩿㠃 2 ḟ࣮ࣔ
࣮࣓ࣥ
ࢺ Ieqq ࠿ࡽᘧ(3.166)࡟ࡼࡗ࡚ồ
ồࡵࡿࡇ࡜ࡀ
ࡀ࡛ࡁࡿ㸬
⥾ࡵ
ࡵ௦ ǻd ࡣ㸪㍈ࡢእᚄ d ࡜ᔐࡵྜࢃ
ࢃࡏࡿࢫ࣮ࣜ
࣮ࣈࡢෆᚄ ds ࡜ࡢᕪ࡛࠶
࠶ࡿ㸦ᘧ㸦4.2㸧ཧ
↷㸧㸬
'd
d d s (4.2)
(
௨ୖ
ୖࡢ‽ഛࡢୗ
ୗ㸪ᦶ᧿ಀᩘ
ᩘ㸪⥾ࡵ௦ࢆ
ࢆゎᯒࣃ࣓ࣛ
࣓࣮ࢱ࡜ࡋ࡚
࡚㸪ᔐࡵྜ࠸
࠸㒊ࡢ᭤ࡆ๛
๛ᛶࢆ
ゎᯒ࡟ࡼࡗ࡚ホ౯
౯ࡋࡓ㸬ࡲࡓ
ࡓ㸪ṇ㈇ࡢ᭤
᭤ࡆ࣮࣓ࣔࣥ
ࣥࢺࢆ⧞ࡾ㏉
㏉ࡋస⏝ࡉࡏ
ࡏࡓሙྜࡢᔐ
ᔐࡵྜ
ࡢ᭤ࡆ๛ᛶ࡟
࡟ࡘ࠸࡚ࡶゎ
ゎᯒ࡟ࡼࡗ࡚
࡚ホ౯ࡋࡓ㸬
࠸㒊ࡢ
49
(1) ᦶ
ᦶ᧿ಀᩘࢆゎ
ゎᯒࣃ࣓࣮ࣛ
࣮ࢱ࡜ࡋࡓゎ
ゎᯒ
⥾ࡵ
ࡵ௦ ǻd ࢆ┤
┤ᚄ࡛ 0.024m
mm ࡢ୍ᐃ್
್࡜ࡋ㸪ᦶ᧿
᧿ಀᩘ ȝ ࢆ 0..05㸪0.2㸪0.55 ࡢ 3 ㏻ࡾ࡟
࡟ኚ໬
ࡉࡏࡓ
ࡓሙྜࡢ㸪᭤
᭤ࡆ࣮࣓ࣔࣥ
ࣥࢺ M ࡜➼౯
౯┤ᚄ deq ࡢ㛵ಀࢆᅗ
ࡢ
4.3 ࡟♧ࡍ㸬ྠ
ྠᅗ࡟ࡣ㸪ᔐ
ᔐࡵྜ
࠸㒊ࡢ
ࡢ᭱ᑠ➼౯㍈
㍈ᚄ 25.4mm
m ࡜᭱኱➼౯
౯㍈ᚄ 35.4mm
m ࢆࡑࢀࡒࢀ
ࢀ㉥Ⰽࡢ◚⥺
⥺࡜ᐇ⥺࡛♧
♧ࡋࡓ㸬
㟼ゎᯒ
ᯒ࡛ࡣ㸪ᦶ᧿
᧿ಀᩘࡀ኱ࡁ
ࡁ࠸࡯࡝ᔐࡵ
ࡵྜ࠸㒊ࡢ➼
➼౯㍈ᚄࡀ኱
኱ࡁ࡞್࡜࡞
࡞ࡿ⤖ᯝࡀᚓ
ᚓࡽࢀ
ࡓ㸬
F
Fig. 4.3
Equuivalent diam
meter versus bbending moment in case off interferencee ǻd=0.024mm
m
(friction coeffficient ȝ=0.05
5, 0.2, and 0.5
5)
⥾ࡵ௦ࢆゎᯒ
ᯒࣃ࣓࣮ࣛࢱ
ࢱ࡜ࡋࡓゎᯒ
ᯒ
(2) ⥾
ᦶ᧿
᧿ಀᩘ ȝ ࢆ 0.1 ࡢ୍ᐃ್
್࡜ࡋ㸪⥾ࡵ
ࡵ௦ ǻd ࢆ 0.0
01mm㸪0.024
4mm㸪0.041m
mm ࡢ 3 ㏻ࡾ
ࡾ࡟ኚ
໬ࡉࡏ
ࡏࡓሙྜࡢ᭤
᭤ࡆ࣮࣓ࣔࣥ
ࣥࢺ࡜➼౯┤
┤ᚄ deq ࡢ㛵ಀࢆᅗ 4.4 ࡟♧ࡍ㸬ྠᅗ
࡟
ᅗ࡟ࡣ㸪ᔐࡵ
ࡵྜ࠸
㒊ࡢ᭱
᭱ᑠ➼౯㍈ᚄ
ᚄ 25.4mm ࡜᭱኱➼౯㍈
࡜
㍈ᚄ 35.4mm
m ࢆࡑࢀࡒࢀ
ࢀ㉥Ⰽࡢ◚⥺
⥺࡜ᐇ⥺࡛♧ࡋࡓ㸬
㟼ゎᯒ
ᯒ࡛ࡣ㸪⥾ࡵ
ࡵ௦࡟ࡼࡗ࡚
࡚ᔐࡵྜ࠸㒊
㒊ࡢ➼౯㍈ᚄ
ᚄࡣṤ࡝ኚ໬
໬ࡋ࡞࠸⤖ᯝ
ᯝࡀᚓࡽࢀࡓ㸬
50
F
Fig. 4.4
Eqquivalent diam
meter versus bbending mom
ment in case of
o friction coeefficient ȝ=0.1
(in
nterference ǻd =0.01, 0.024
4, and 0.041m
mm)
(3) ṇ
ṇ㈇ࡢ⧞ࡾ㏉
㏉ࡋ᭤ࡆ࣮ࣔ
࣮࣓ࣥࢺࢆ୚
୚࠼ࡓゎᯒ
ᦶ᧿
᧿ಀᩘ ȝ ࢆ 0.1㸪⥾ࡵ௦
௦ ǻd ࢆ 0.01m
mm ࡜ࡋ࡚㸪ᅗ 4.5 ࡟♧
♧ࡋࡓ±200Nm
mm ࡢ᭤ࡆࣔ
࣮࣓ࣔ
ࣥࢺࢆ⧞ࡾ㏉ࡋ㈇
㈇Ⲵࡋ㸪ኚ఩
఩ࡢᒚṔࢆㄪ
ㄪ࡭ࡓ㸬᭤ࡆ
ࡆ࣮࣓ࣔࣥࢺ
ࢺ࡜┦ᑐࡓࢃ
ࢃࡳ į0-į1㸦ᅗ
ᅗ 4.2
㸧ࡢ㛵ಀࢆᅗ
ᅗ 4.6 ࡟♧ࡍ
ࡍ㸬ྠᅗࡼࡾ 㸪1 ᅇ┠ࡢࢧ
ࢧ࢖ࢡࣝ࡜ 2 ᅇ┠௨㝆ࡢ
ࡢࢧ࢖ࢡ࡛ࣝ
࡛ࡳ࠿
ཧ↷㸧
ࡅࡢ᭤ࡆ๛ᛶࡀኚ
ኚ໬ࡋ㸪ࢮࣟ
ࣟⅬ㸦Ⲵ㔜 0 ࡛ࡢኚ఩㸧࡟ࡎࢀࡀ⏕
⏕ࡌࡿࡇ࡜ࡀ
ࡀศ࠿ࡗࡓ㸬
ն
ղ
ձ
շ
յ
ճ
մ
Fig. 4.5
պ
չ
ո
Allternating ben
nding momen
nt
51
ջ
ռ
ղ
նպ
2nd Cycle
1st Cycle
յ
ձ չջշ
ճ
մ ոռ
Fig. 4.6
Beending moment versus relaative deflectio
on in case of interference ǻ d=0.024mm
m
and fricttion coefficieent ȝ=0.1
᭤ࡆ
ࡆ࣮࣓ࣔࣥࢺ࡜┦ᑐࡓࢃ
ࢃࡳ į0-į1 ࡢ 㛵ಀࡀ཰᮰ࡋࡓ࡜ࡇࢁ࡛
࡛ࢮࣟⅬࢆ⿵
⿵ṇࡋ࡚ồࡵ
ࡵࡓ㸪
᭤ࡆࣔ
࣮࣓ࣔࣥࢺ࡜
࡜➼౯┤ᚄࡢ
ࡢ㛵ಀࢆᅗ 44.7 ࡟♧ࡍ㸬ྠᅗ࡟ࡣ㸪ᦶ᧿ಀᩘ ȝ ࢆ 0.2 ࡜ࡋࡓ
ࡓሙྜ
࡜㸪ࢫ
ࢫ࣮ࣜࣈࡀ㍈
㍈࡜୍య࡜࡞
࡞ࡗࡓࢯࣜࢵ
ࢵࢻࣔࢹࣝࡢ
ࡢሙྜࡢ᭤ࡆ
ࡆ࣮࣓ࣔࣥࢺ
ࢺ࡜➼౯┤ᚄ
ᚄࡢ㛵
ಀ࡟ࡘ
ࡘ࠸࡚ࡶ♧ࡋ
ࡋࡓ㸬ȝ=0.1 ࡜ ȝ=0.2 ࡢ ሙྜࡢ➼౯㍈
㍈ᚄࡣ㸪࡝ࡕ
ࡕࡽࡶⲴ㔜ࡢ
ࡢ್࡟ࡼࡗ࡚
࡚ኚ໬
ࡏࡎ㸪
㸪ࢯࣜࢵࢻࣔ
ࣔࢹࣝࡢࡶࡢ
ࡢ࡜ྠ୍ࡢ 335.09mm㸦᭱
᭱኱➼౯㍈ᚄ
ᚄࡢ 99.12%㸧 ࡜࠸࠺್࡜
࡜࡞ࡗ
ࡓ㸬
F
Fig. 4.7
Equuivalent diam
meter versus bbending moment in case off interferencee ǻd=0.001mm
m
(friction coefficieent ȝ=0.1 and
d 0.2, solid model)
52
ࡾ⠊ᅖศᕸᅗ
ᅗࢆ♧ࡍ㸬ࡍ
ࡍ࡭ࡾ࡞ࡋࡢ㡿ᇦࡀ㉥Ⰽ㸪
㸪ࡍ࡭ࡾ࠶ࡾ
ࡾࡢ㡿ᇦࡀ㟷
㟷Ⰽ࡛
ᅗ 4.8 ࡟ࡍ࡭ࡾ
ࢀ࡚࠸ࡿ㸬ྠ
ྠᅗ࡟࠾ࡅࡿ
ࡿྑ➃᩿㠃ࡢ
ࡢእ࿘㏆ഐࡢ
ࡢ╔Ⰽࡉࢀࡓ
ࡓ㒊ศࡣ㸪ゎ
ゎᯒࣉࣟࢢࣛ
࣒ࣛࡢ
♧ࡉࢀ
⾲♧ୖ
ୖࡢၥ㢟࡟ࡼࡿࡶࡢ࡛࠶
࠶ࡾ㸪ࡍ࡭ࡾ
ࡾ࡜ࡣ↓㛵ಀ
ಀ࡛࠶ࡿ㸬ࡇ
ࡇࡢศᕸᅗ࠿
࠿ࡽ㸪᭤ࡆࣔ
࣮࣓ࣔ
ࣥࢺࡢ
࡟కࡗ࡚ࡍ࡭
࡭ࡾ⠊ᅖࡀᑠ
ᑠࡉࡃ࡞ࡾ㸪࡯ࡰࡍ࡭࡚
࡚ࡢ㡿ᇦ࡛⁥
⁥ࡾࡀ࡞ࡃ࡞
࡞ࡿ≧
ࡢ⧞ࡾ㏉ࡋ࡟
ែ࡟཰
཰᮰ࡋ࡚࠸ࡿ
ࡿࡇ࡜ࡀศ࠿
࠿ࡿ㸬
S
Step 1 (M=0 Nmm)
Step 2 ((M=200 Nmm
m)
Step
S 3 (M=0 N
Nmm)
S
Step 4 (M=-2200 Nmm)
Step 5 (M=0 Nmm
m)
Step 6 (M=2000 Nmm)
Step 7 (M=00 Nmm)
Step 8 (M=-200 Nm
mm)
Step 10 (M=
=200 Nmm)
Fiig. 4.8
Step 11 (M=0 Nm
mm)
Step 9 (M=0 Nmm)
Step 12 (M==-200 Nmm)
Slidding area map
p in case of innterference ǻd=0.01mm and
a friction cooefficient ȝ=0
0.1
53
᭤ࡆ࣮࣓ࣔࣥࢺࢆ⧞ࡾ㏉ࡋ㈇Ⲵࡋࡓሙྜࡢᔐࡵྜ࠸㒊ࡢ≧ែࡣ㸪᭤ࡆ࣮࣓ࣔࣥࢺࡢ኱ࡁ
ࡉ࡟ࡼࡗ࡚␗࡞ࡿࡓࡵ㸪ᅗ 4.8 ࡢ᭤ࡆ࣮࣓ࣔࣥࢺࡀᑠࡉ࠸ሙྜ࡜ᚋ㏙ࡍࡿ᭤ࡆ࣮࣓ࣔࣥࢺࡀ
኱ࡁ࠸ሙྜ࡟ศࡅ࡚ㄝ᫂ࡍࡿ㸬
(a) ࢫࢸࢵࣉ 1 ࡣ㸪㍈࡟᭤ࡆ࣮࣓ࣔࣥࢺࡀస⏝ࡋ࡚࠸࡞࠸㍈࡟㸪㍈ࡢእᚄࡼࡾᑠࡉ࠸ෆᚄ
ࢆᣢࡘࢫ࣮ࣜࣈࢆᔐࡵྜࢃࡏࡓ≧ែ࡛࠶ࡾ㸪ࢫ࣮ࣜࣈࡢෆᚄࡀᣑࡆࡽࢀࡿࡢ࡟కࡗ࡚
ࢫ࣮ࣜࣈࡀ㍈᪉ྥ࡟ᅽ⦰ࡉࢀࡿࡓࡵࡍ࡭ࡾࡀ⏕ࡌ࡚࠸ࡿ㸬ᦶ᧿ຊ q ࡣࢫ࣮ࣜࣈࡢ⦰ࡳ
ࢆ⦆࿴ࡍࡿ᪉ྥ࡟Ⓨ⏕ࡋ㸪ࡑࡢ኱ࡁࡉ|q|ࡣᘧ㸦4.3㸧࡛⾲ࡉࢀࡿ㸬
q
Pp 㸦4.3㸧
ȝ㸸ᦶ᧿ಀᩘ
p㸸㠃ᅽⲴ㔜 (N)
ࡇࡇ࡛㸪㠃ᅽⲴ㔜 p ࡣ㸪㍈࡟స⏝ࡍࡿ᭤ࡆ࣮࣓ࣔࣥࢺ࡟౫Ꮡࡍࡿ㸪ᔐࡵྜ࠸㒊ࡢ㍈ࡢ
ᙎᛶኚᙧࡢᙧ≧࡟ࡼࡗ࡚ኚ໬ࡍࡿࡓࡵ㸪ᘧ㸦4.4㸧࡛⾲ࡍࡇ࡜ࡀ࡛ࡁࡿ㸬
p
p0 'p 㸦4.4㸧
p0㸸ᔐࡵྜ࠸࡟ࡼࡿ㠃ᅽⲴ㔜 (N)
ǻp㸸᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ㠃ᅽⲴ㔜ࡢኚ໬ศ (N)
ǻp ࡣ㸪㍈ࡀฝ࡜࡞ࡿഃࡢࢫ࣮ࣜࣈ➃㒊࡛ࡣ㈇࡜࡞ࡾ㸪㍈ࡀพ࡜࡞ࡿഃࡢࢫ࣮ࣜࣈ➃
㒊࡛ࡣṇ࡜࡞ࡿ㸬ᘧ㸦4.3㸧࡜ᘧ㸦4.4㸧ࡼࡾ㸪ᦶ᧿ຊ q ࡢ኱ࡁࡉ|q|ࡣᘧ㸦4.5㸧࡛⾲ࡉ
ࢀࡿ㸬
q
P p0 'p 㸦4.5㸧
ࢫࢸࢵࣉ 1 ࡛ࡣ᭤ࡆ࣮࣓ࣔࣥࢺࡀస⏝ࡋ࡚࠸࡞࠸ࡓࡵ㸪ᦶ᧿ຊ q ࡢ኱ࡁࡉ|q|ࡣ ȝp0 ࡛
࠶ࡿ㸬
(b) ᭤ࡆ࣮࣓ࣔࣥࢺࡀᑠࡉ࠸ሙྜࡣ㸪ࢫࢸࢵࣉ 2 ࡛ࡣ㸪㍈ࡢฝഃ㸦ୗ㠃㸧ࡢࢫ࣮ࣜࣈ➃㒊
ࡢ㠃ᅽࡀపୗࡍࡿࡢ࡟క࠸㍈ࡀ㍈᪉ྥ࡟ఙࡧࡿࡓࡵ㸪ᦶ᧿ຊࡀ᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ
㍈ࡢᘬᙇ᪉ྥࡢࡏࢇ᩿ຊ࡜ྠ୍᪉ྥ࡟స⏝ࡋ࡚ఙࡧ᪉ྥࡢࡍ࡭ࡾࡀ⏕ࡌ㸪㍈ࡢพഃ
㸦ୖ㠃㸧ࡢࢫ࣮ࣜࣈ➃㒊࡛ࡣ᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ㍈ࡢᅽ⦰᪉ྥࡢࡏࢇ᩿ຊࡀ㠃ᅽୖ
᪼࡟కࡗ࡚ቑ኱ࡋࡓ㝈⏺ᦶ᧿ຊࢆୗᅇࡿࡓࡵࡍ࡭ࡾࡣ⏕ࡌ࡞࠸㸬ࢫࢸࢵࣉ 3 ࡛ࡣ㸪ࢫ
ࢸࢵࣉ 2 ࡛ୗ㠃࡟⏕ࡌࡓࡍ࡭ࡾࡀಖᣢࡉࢀ㸪ୖ㠃࡛ࡣࡍ࡭ࡾ↓ࡋࡢ≧ែࡀಖᣢࡉࢀࡿ
ࡓࡵ㸪᭤ࡆࡢ୰❧㠃࡟㏆࠸ഃ㠃ࡣࢫࢸࢵࣉ 1 ࠿ࡽ≧ែࡀኚ໬ࡋ࡞࠸㸬ࢫࢸࢵࣉ 4 ࡛ࡣ㸪
54
ࢫࢸࢵࣉ 2 ࡜ྠᵝ࡟㸪㍈ࡢฝഃ㸦ୖ㠃㸧࡛ࡢࡳఙࡧ᪉ྥࡢࡍ࡭ࡾࡀ⏕ࡌࡿ㸬ࢫࢸࢵࣉ
5 ࡛ࡣ㸪ࢫࢸࢵࣉ 3 ࡜ྠᵝ࡟ୖ㠃࡜ୗ㠃࡟ࡍ࡭ࡾࡣ⏕ࡌ࡞࠸㸬ࢫࢸࢵࣉ 6 ௨㝆ࡣ㸪㠃
ᅽࡀపୗࡍࡿ㍈ࡢฝഃ࡟࠾࠸࡚ࡶ㸪᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ㍈ࡢᘬᙇ᪉ྥࡢࡏࢇ᩿ຊࡀ
㝈⏺ᦶ᧿ຊࢆୖᅇࡽ࡞࠸ࡓࡵ㸪ୖୗ㠃ඹࡍ࡭ࡾࡀ⏕ࡌ࡞࠸≧ែ࡜࡞ࡿ㸬
(c) ᭤ࡆ࣮࣓ࣔࣥࢺࡀ኱ࡁ࠸ሙྜࡣ㸪ࢫࢸࢵࣉ 2 ࡛ࡣ㸪᭤ࡆ࣮࣓ࣔࣥࢺࡀᑠࡉ࠸ሙྜ࡜ྠ
ᵝ࡟㸪㍈ࡢฝഃ㸦ୗ㠃㸧ࡢࢫ࣮ࣜࣈ➃㒊ࡢ㠃ᅽࡀపୗࡍࡿࡢ࡟క࠸ࢫ࣮ࣜࣈ➃㒊ࡢෆ
ᚄഃࡀ㍈᪉ྥ࡟⦰ࡶ࠺࡜ࡍࡿࡓࡵ㸪ᦶ᧿ຊࡀ᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ㍈ࡢᘬᙇ᪉ྥࡢࡏ
ࢇ᩿ຊ࡜ྠ୍᪉ྥ࡟స⏝ࡋ࡚ఙࡧ᪉ྥࡢࡍ࡭ࡾࡀ⏕ࡌࡿ㸬㍈ࡢพഃ㸦ୖ㠃㸧ࡢࢫ࣮ࣜ
ࣈ➃㒊࡛ࡣ᭤ࡆ࣮࣓ࣔࣥࢺ࡟ࡼࡿ㍈ࡢᅽ⦰᪉ྥࡢࡏࢇ᩿ຊࡀ㸪㠃ᅽୖ᪼࡟కࡗ࡚ቑ኱
ࡋࡓ㝈⏺ᦶ᧿ຊࢆୖᅇࡾ⦰ࡳ᪉ྥࡢࡍ࡭ࡾࡀ⏕ࡌࡿ㸬ࢫࢸࢵࣉ 3 ࡛ࡣ㸪ୗ㠃࡟⦰ࡳ᪉
ྥࡢࡍ࡭ࡾࡀ⏕ࡌࡿ㸬ࢫࢸࢵࣉ 4 ࡛ࡣ㸪㍈ࡢฝഃ㸦ୖ㠃㸧࡛ࡣఙࡧ᪉ྥࡢࡍ࡭ࡾࡀ⏕
ࡌ㸪㍈ࡢพഃ㸦ୗ㠃㸧࡛ࡣ⦰ࡳ᪉ྥࡢࡍ࡭ࡾࡀ⏕ࡌࡿ㸬ࢫࢸࢵࣉ 5 ࡛ࡣ㸪ୖ㠃࡟⦰ࡳ
᪉ྥࡢࡍ࡭ࡾࡀ⏕ࡌࡿ㸬ࢫࢸࢵࣉ 6 ௨㝆ࡣ㸪ࢫࢸࢵࣉ 2 ࠿ࡽࢫࢸࢵࣉ 5 ࡜ྠᵝ࡟㸪㍈
ࡢฝഃࡢࢫ࣮ࣜࣈ➃㒊࡛ࡣ㸪᭤ࡆ࣮࣓ࣔࣥࢺࡢ኱ࡁࡉࡀቑ኱ࡍࡿ㐣⛬࡛ࡣఙࡧ᪉ྥࡢ
ࡍ࡭ࡾࡀ⏕ࡌ㸪ῶᑡࡍࡿ㐣⛬࡛ࡣ⦰ࡳ᪉ྥࡢࡍ࡭ࡾࡀ⏕ࡌࡿ㸬㍈ࡢพഃࡢࢫ࣮ࣜࣈ➃
㒊࡛ࡣ㸪㍈ࡢฝഃ࡜㏫ྥࡁࡢࡍ࡭ࡾࡀ⏕ࡌࡿ㸬
ḟ࡟㸪⧞ࡾ㏉ࡋస⏝ࡍࡿ᭤ࡆ࣮࣓ࣔࣥࢺࡀ኱ࡁ࠸ሙྜ࡟ࡘ࠸࡚㸪⁥ࡾࡀ⏕ࡌࡿሙྜࡢぢ
࠿ࡅࡢ᭤ࡆ๛ᛶ࡟ࡘ࠸᳨࡚ウࡍࡿ㸬౛࡜ࡋ࡚㸪ᅗ 4.1 ࡟࠾࠸࡚ᔐࡵྜࢃࡏࡿࢫ࣮ࣜࣈࡢཌࡉ
ࢆ 1.0mm ࡜ࡋ࡚ᔐࡵྜ࠸࡟ࡼࡿ㠃ᅽⲴ㔜 p0 ࢆᑠࡉࡃࡋ㸪ᦶ᧿ಀᩘ ȝ ࢆ 0.05㸪⥾ࡵ௦ ǻd ࢆ
0.01mm㸪㈇Ⲵࡍࡿ⧞ࡾ㏉ࡋࡢ᭤ࡆ࣮࣓ࣔࣥࢺࢆᅗ 4.5 ࡢ 100 ಸࡢ±20000Nmm ࡜ࡋࡓሙྜࡢ㸪
᭤ࡆ࣮࣓ࣔࣥࢺ࡜┦ᑐࡓࢃࡳ į0-į1㸦ᅗ 4.2 ཧ↷㸧ࡢ㛵ಀࢆᅗ 4.9 ࡟♧ࡍ㸬ྠᅗࡼࡾ㸪⁥ࡾࡀ
⏕ࡌࡿሙྜࡣ㸪1 ᅇ┠ࡢ 1/4 ࢧ࢖ࢡࣝ௨㝆ࡣྠ୍ࡢࣄࢫࢸࣜࢩࢫ࣮ࣝࣉࢆᥥ࠸࡚ࡳ࠿ࡅࡢ᭤
ࡆ๛ᛶࡀኚ໬ࡏࡎ㸪ࡳ࠿ࡅࡢ᭤ࡆ๛ᛶࡣᅗ 4.3 ࡟♧ࡉࢀࡓᦶ᧿ಀᩘ࡟౫Ꮡࡍࡿ್࡜࡯ࡰྠ୍
࡜࡞ࡿࡇ࡜ࡀศ࠿ࡗࡓ㸬
55
2nd Cyclle
Fig. 4.9
1st Cycle
Beending momeent versus rellative deflectiion in case off interferencee ǻd=0.01mm
m
and fricttion coefficiennt ȝ=0.1 (sleeeve thickness: 1.0mm)
௨ୖ
ୖࡢ᳨ウࡢ⤖
⤖ᯝ㸪⥾ࡵ௦
௦ ǻd ࡀṇ࡛ ࠶ࢀࡤ㸪㠃ᅽ
ᅽⲴ㔜 p ࡟ᑐ
ᑐࡋ࡚᭤ࡆࣔ
࣮࣓ࣔࣥࢺ M ࡀᑠ
ࡉ࠸ሙ
ሙྜࡣ㸪ᦶ᧿
᧿ಀᩘ ȝ ࡀᑠࡉ࡞್࡛
ࡀ
ࡶࡍ࡭ࡾࡀ⏕
⏕ࡌ࡞࠸ࡓࡵ
ࡵ᭤ࡆ๛ᛶࡀ
ࡀ➼౯࡜࡞ࡿ
ࡿ㍈ᚄ
ࡣࢫ࣮ࣜࣈእᚄ࡟
࡟㏆࠸್࡜࡞
࡞ࡾ㸪㠃ᅽ࡟
࡟ᑐࡋ࡚᭤ࡆ
ࡆ࣮࣓ࣔࣥࢺ
ࢺ M ࡀ኱ࡁ࠸
࠸ሙྜࡣ㸪ᦶ
ᦶ᧿ಀ
࡜⁥ࡾࡀ⏕ࡌ
ࡌࡿࡓࡵ᭤ࡆ ๛ᛶࡀ➼౯࡜࡞ࡿ㍈ᚄࡣ
ࡣᦶ᧿ಀᩘ࡟
࡟౫Ꮡࡋࡓࢫ
ࢫ࣮ࣜ
ᩘ ȝ ࡀᑠࡉ࠸࡜
ࣈእᚄ
ᚄࡼࡾᑠࡉ࡞
࡞್࡜࡞ࡿࡇ
ࡇ࡜ࡀ☜ㄆࡉ
ࡉࢀࡓ㸬
ᔐࡵྜ࠸
࠸㒊ࡢ᭤ࡆ๛
๛ᛶࡢᐇ㦂࡟ ࡼࡿホ౯
4.22.1 ⠇ࡢ᥋ゐ
ゐゎᯒ࡟ࡼࡿ᳨ウ࡟ࡼࡗ ࡚㸪ᔐࡵྜ࠸
࠸㒊ࢆᣢࡘ㍈
㍈࡟࠾࠸࡚㸪 ⥾ࡵ௦ࡀṇ
ṇ࡛࠶
ࢀࡤ᭤ࡆ๛ᛶࡀ➼
➼౯࡜࡞ࡿ㍈
㍈ᚄࡣࢫ࣮ࣜ
࣮ࣈእᚄ࡟࡯
࡯ࡰ୍⮴ࡍࡿ
ࡿ࡜࠸࠺⤖ᯝ
ᯝࡀᚓࡽࢀࡓ
ࡓ㸬ࡲ
⥾ࡵ௦ࡀṇ࡛
࡛࠶ࢀࡤ᭤ࡆ
ࡆ๛ᛶࡀ➼౯
౯࡜࡞ࡿ㍈ᚄ
ᚄࡣ㍈ࡢእᚄ
ᚄ࡟୍⮴ࡍࡿ
ࡿ࡜᥎ ࡉࢀ
ࢀࡿ㸬
ࡓ㸪⥾
ࡑࡇ࡛
࡛㸪ࢫ࣮ࣜࣈ
ࣈࢆࡋࡲࡾࡤ
ࡤࡵ࡛ᔐࡵྜ
ྜࢃࡏࡓ㍈࡟
࡟ࡘ࠸࡚㸪3.4.4 ⠇࡜ྠᵝ
ᵝࡢ⾪ᧁຍ᣺
᣺ヨ㦂
ࢆᐇ᪋
᪋ࡋ࡚⮬⏤ᨭ
ᨭᣢ≧ែࡢ㍈
㍈ࡢ᭤ࡆ࣮ࣔ
࣮ࢻࡢᅛ᭷᣺
᣺ືᩘࢆồࡵ
ࡵ㸪➨ 3 ❶࡛
࡛ᥦ᱌ࡋࡓ㍈
㍈┤ᚄ
ኚ໬㒊
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧࢆ⪃៖
៖ࡋࡓ 1D FE
EM ࣔࢹ࡛ࣝ
࡛ồࡵࡓᅛ᭷
᭷᣺ືᩘ࡜ẚ
ẚ㍑ࡋ࡚㸪ᔐ
ᔐࡵྜ
࠸㒊ࡢ
ࡢ➼౯᭤ࡆ๛
๛ᛶࢆホ౯ࡋ
ࡋࡓ㸬
ࢫ࣮ࣜࣈࢆᔐࡵ
ࡵྜࢃࡏࡿ㍈
㍈ࡣ㸪ᅗ 3.133 (a)࡟♧ࡋࡓ
ࡓ㍈┤ᚄኚ໬
໬㒊ࢆᣢࡘ㍈
㍈ A ࡜ࡋࡓ㸬ࢫࣜ
ࡢ㍈᪉ྥ㛗ࡉ
ࡉࡣ 25.4 mm
m㸪ࢫ࣮ࣜࣈࡢ
ࡢཌࡉࡣ 5 mm
m 㸪ᔐࡵྜ࠸
࠸ᙉࡉࡣࡋࡲ
ࡲࡾࡤࡵ࡜ࡋ
ࡋࡓ㸬
࣮ࣈࡢ
ᅗ 4.10 (a)࡟ࢫ
ࢫ࣮ࣜࣈ௜ࡁࡢ㍈ࡢ 1D F
FEM ࣔࢹࣝ
ࣝࢆ♧ࡋ㸪ᅗ
ᅗ 4.10 (b)࡟ྠ
ྠᅗ(a)ࡢࢫࣜ
࣮ࣜࣈ
㍈࡜ྠ୍ᙧ≧
≧ࡢ୰ᐇ㍈ࡢ
ࡢ 1D FEM ࣔ
ࣔࢹࣝࢆ♧ࡍ
ࡍ㸬ྠᅗ(a)㸪(b)ඹ࡟㸪┤
┤ᚄኚ໬㒊ࡢ
ࡢᙎᛶ
௜ࡁ㍈
ኚᙧࢆ⪃៖ࡋࡓࣔ
ࣔࢹࣝ㸦1D Model 3㸧࡛
࡛࠶ࡿ㸬
56
(a) S
Saft with a slleeve
((b) Solid shafft
F 4.10
Fig.
1D
1 FEM moddels of shaft with
w a sleeve and solid shaaft
㍈┤
┤ᚄኚ໬㒊᩿
᩿㠃ࡢᙎᛶኚ
ኚᙧࢆ⪃៖ࡋ
ࡋࡓ 1D FEM
M ࣔࢹࣝ࡟ࡼ
ࡼࡿᔐࡵྜ࠸
࠸㒊ࡢ➼౯᭤
᭤ࡆ๛
ᛶࡢホ
ホ౯࡟ඛ❧ࡕ
ࡕ㸪1D FEM
M ࣔࢹࣝ࡟ࡼ
ࡼࡗ࡚㸪ᅗ 4.10
4
(a)ࡢࡍࡁ
ࡁࡲࡤࡵࡢࢫ
ࢫ࣮ࣜࣈ௜ࡁ
ࡁ㍈࡜
ᅗ 4.110 (b)ࡢࢯࣜ
ࣜࢵࢻ㍈࡜ࡢ
ࡢᅛ᭷᣺ືᩘ
ᩘࡢᕪ␗ࢆ㸪3D FEM ࣔࢹ
ࢹࣝ࡜ྠᵝ࡟
࡟⾲⌧࡛ࡁࡿ
ࡿࡇ࡜
ࢆ☜ㄆ
ㄆࡍࡿ㸬
ᐇ㦂
㦂࡟ࡼࡿᅗ 4.10 (a)ࡢࡋ
ࡋࡲࡾࡤࡵࢫ ࣮ࣜࣈ௜ࡁ㍈
㍈ࡢᔐࡵྜ࠸
࠸㒊ࡢ᭤ࡆ๛
๛ᛶࡢホ౯࡟
࡟ඛ❧
ࡕ㸪ᅗ
ᅗ 4.10 (b)ࡢࢯࣜࢵࢻ㍈ࡢ᭤ࡆ࣮ࣔ ࢻࡢᅛ᭷᣺ື
ືᩘࡀ㸪ᅗ 4.10
4 (a)ࡢ㍈࡟
࡟࠾࠸࡚ࢫࣜ
࣮ࣜࣈ
ࡢᔐࡵྜ࠸ᙉࡉࢆ
ࢆࡍࡁࡲࡤࡵ
ࡵ࡜ࡋࡓ㍈ࡢ
ࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ື
ືᩘ࡜㸪᫂☜
☜࡟㆑ู࡛ࡁ
ࡁࡿࡇ
☜ㄆࡍࡿࡓࡵ
ࡵ࡟㸪ゎᯒ࡟
࡟ࡼࡿ᳨ウࢆ
ࢆ⾜ࡗࡓ㸬
࡜ࢆ☜
⾲ 4.1 ࡣ㸪ᅗ 4.10
4
(b)ࡢࢯࣜࢵࢻ㍈ࡢ 3 ḟඖࣔࢹࣝ
ࣝ㸦3D Modeel㸧࡜ᅗ 4.100 (a)ࡢࡍࡁࡲ
ࡲࡤࡵ
㍈ࡢ 1 ḟඖࣔ
ࣔࢹࣝ㸦1D M
Model 3㸧࡟ࡘ
ࡘ࠸࡚᭤ࡆࣔ
࣮ࣔࢻࡢᅛ᭷
᭷᣺ືᩘࡢゎ
ゎᯒ್
ࡢࢫ࣮ࣜࣈ௜ࡁ㍈
㍑ࢆ♧ࡋࡓࡶ
ࡶࡢ࡛࠶ࡾ㸪
⾲ 4.2 ࡣ㸪ᅗ 4.10 (b)ࡢࢯ
ࢯࣜࢵࢻ㍈ࡢ
ࡢ 1 ḟඖࣔࢹ
ࢹࣝ㸦1D Mod
del 3㸧
ࡢẚ㍑
࡜ᅗ 4.10 (a)ࡢࡍ
ࡍࡁࡲࡤࡵࡢࢫ࣮ࣜࣈ௜ ࡁ㍈ࡢ 1 ḟඖ
ඖࣔࢹࣝ㸦1D
1 Model 3㸧 ࡟ࡘ࠸࡚᭤
᭤ࡆࣔ
ࡢᅛ᭷᣺ືᩘ
ᩘࡢゎᯒ್ࡢ
ࡢẚ㍑ࢆ♧ࡋ
ࡋࡓࡶࡢ࡛࠶
࠶ࡿ㸬ࡍࡁࡲ
ࡲࡤࡵࡢࢫࣜ ࣮ࣈ௜ࡁ㍈ࡢ 1D
࣮ࢻࡢ
Modeel 3 ࡟ࡼࡿ 1 ḟ࡜ 3 ḟࡢ
ࡢ᭤ࡆ࣮ࣔࢻ
ࢻࡢᅛ᭷᣺ືᩘࡢゎᯒ್ࡣ㸪ࢯࣜࢵ ࢻ㍈ࡢ 3D Model
M
㸪
1D M
Model 3 ࡢఱࢀ
ࢀ࡜ẚ࡭࡚ࡶ
ࡶ 1%௨ୖప
ప࠸್࡜࡞ࡗ
ࡗ࡚࠾ࡾ㸪ࢯࣜࢵࢻ㍈࡜ࡍ
ࡍࡁࡲࡤࡵࡢ
ࡢࢫࣜ
࣮ࣈ௜
௜ࡁ㍈࡛ࡣ 1 ḟ࡜ 3 ḟࡢ
ࡢ᭤ࡆ࣮ࣔࢻ
ࢻࡢᅛ᭷᣺ື
ືᩘࡀ᫂☜࡟
࡟␗࡞ࡿࡇ࡜ ࡀศ࠿ࡿ㸬
57
Table 4.1 Comparison between analytical natural frequencies of 3D Model of solid shaft and
analytical natural frequencies of 1D Model 3 of shaft with a sleeve set on it by clearance fit
Unit: Hz
Analytical value of 1D
Analytical
Model 3 of shaft with a
value of 3D
Mode
sleeve on it by clearance
Model of
fit (difference)
solid shaft
1
733.33
724.34
(-1.23%)
2
2034.8
2038.2
(0.17%)
3
3804.9
3753.9
(-1.34%)
4
6161.8
6174.5
(0.21%)
Table 4.2 Comparison between analytical natural frequencies of 1D Model 3 of solid shaft and
analytical natural frequencies of 1D Model 3 of shaft with a sleeve set on it by clearance fit
Unit: Hz
Analytical value of 1D
Analytical
Model 3 of shaft with a
value of 1D
Mode
sleeve on it by clearance
Model 3 of
fit (difference)
solid shaft
1
731.99
724.34
(-1.05%)
2
2036.1
2038.2
(0.10%)
3
3812.8
3753.9
(-1.54%)
4
6160.2
6174.5
(0.23%)
⾲ 4.3 ࡣ㸪ᅗ 4.10 (b)ࡢࢯࣜࢵࢻ㍈ࡢ 3 ḟඖࣔࢹࣝ㸦3D Model㸧ࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ື
ᩘࡢゎᯒ್࡜ᅗ 4.10 (a)ࡢࡋࡲࡾࡤࡵࡢࢫ࣮ࣜࣈ௜ࡁ㍈ࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡢᐇ㦂್
ࡢẚ㍑ࢆ♧ࡋࡓࡶࡢ࡛࠶ࡾ㸪⾲ 4.4 ࡣ㸪ᅗ 4.10 (b)ࡢࢯࣜࢵࢻ㍈ࡢ 1 ḟඖࣔࢹࣝ㸦1D Model 3㸧
ࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡢゎᯒ್࡜ᅗ 4.10 (a)ࡢࡋࡲࡾࡤࡵࡢࢫ࣮ࣜࣈ௜ࡁ㍈ࡢ᭤ࡆ࣮ࣔ
ࢻࡢᅛ᭷᣺ືᩘࡢᐇ㦂್ࡢẚ㍑ࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬ࢯࣜࢵࢻ㍈ࡢ 3D Model ࠾ࡼࡧ 1D
Model 3 ࡢ 1 ḟ㹼4 ḟࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡢゎᯒ್࡜㸪ࡋࡲࡾࡤࡵࡢࢫ࣮ࣜࣈ௜ࡁ㍈
ࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡢᐇ㦂್࡜ࡢᕪ␗ࡣ㸪ࡑࢀࡒࢀ 0.2%௨ୗ࠾ࡼࡧ 0.4%௨ୗ࡜ᑠࡉ
࡞್࡜࡞ࡗ࡚࠾ࡾ㸪ୖ㏙ࡋࡓ㸪ゎᯒ࡟ࡼࡿࡍࡁࡲࡤࡵࡢࢫ࣮ࣜࣈ௜ࡁ㍈࡜ࡢẚ㍑࡛☜ㄆࡉ
ࢀࡓ࣮ࣔࢻḟᩘ࡟ࡼࡿ㢧ⴭ࡞ᕪ␗ࡶㄆࡵࡽࢀ࡞࠸㸬ࡼࡗ࡚㸪ࡋࡲࡾࡤࡵࡢࢫ࣮ࣜࣈ௜ࡁ㍈
ࡢᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࡣ㸪ࢫ࣮ࣜࣈእᚄ࡜ྠ୍ᚄࡢࢯࣜࢵࢻ㍈࡜࡯ࡰྠ୍࡜ᢅࡗ࡚Ⰻ࠸
࡜ゝ࠼ࡿ㸬
58
Table 4.3 Comparison between analytical natural frequencies of 3D Model of solid shaft and
measured natural frequencies of shaft with a sleeve set on it by interference fit
Unit: Hz
Measured value of
Analytical
shaft with a sleeve
value of 3D
Mode
set on it by shrink fit
Model of
(Error)
solid shaft
1
733.33
731.84
(-0.20%)
2
2034.8
2035.1
(0.01%)
3
3804.9
3797.4
(-0.20%)
4
6161.8
6160.2
(-0.20%)
Table 4.4 Comparison between analytical natural frequencies of 1D Model 3 of solid shaft and
measured natural frequencies of shaft with a sleeve set on it by interference fit
Unit: Hz
Measured value of
Analytical
shaft with a sleeve
value of 1D
Mode
set on it by shrink fit
Model 3 of
(Error)
solid shaft
1
731.99
731.84
(-0.02%)
2
2036.1
2035.1
(-0.05%)
3
3812.8
3797.4
(-0.40%)
4
6160.2
6160.2
(0.00%)
⤖ゝ
ᔐࡵྜ࠸࡟ࡼࡗ࡚㍈ࡢ᭤ࡆ๛ᛶࡀ࡝ࡢ⛬ᗘቑ኱ࡍࡿ࠿ࢆᢕᥱࡍࡿࡓࡵ㸪ࢫ࣮ࣜࣈࡢᔐࡵ
ྜ࠸㒊ࡢᦶ᧿ࢆ⪃៖ࡋࡓ㠀⥺ᙧ㟼ゎᯒ࡜㸪ࢫ࣮ࣜࣈࢆࡋࡲࡾࡤࡵ࡛ᔐࡵྜࢃࡏࡓ㍈࡟ࡘ࠸
࡚⮬⏤ᨭᣢ≧ែࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘィ ࢆᐇ᪋ࡋࡓ㸬
ゎᯒࡢ⤖ᯝ㸪ᦶ᧿ಀᩘ࡜⥾ࡵ௦ࡀᑠࡉ࠸ሙྜ࡛ࡶ⥾ࡵ௦ࡀṇ࡛࠶ࢀࡤ㸪㠃ᅽⲴ㔜࡟ᑐࡋ
࡚᭤ࡆ࣮࣓ࣔࣥࢺࡀᑠࡉ࠸ሙྜࡣᦶ᧿ಀᩘࡀᑠࡉ࡞್࡛ࡶࡍ࡭ࡾࡀ⏕ࡌ࡞࠸ࡓࡵ㸪᭤ࡆ๛
ᛶࡀ➼౯࡜࡞ࡿ㍈ᚄࡀ࡯ࡰࢫ࣮ࣜࣈእᚄ࡟୍⮴ࡍࡿࡇ࡜ࡀ☜ㄆࡉࢀ㸪ᐇ㦂࡟࠾࠸࡚ࡶ㸪ࢫ
࣮ࣜࣈࢆࡋࡲࡾࡤࡵ࡛ᔐࡵྜࢃࡏࡓ㍈ࡢ᭤ࡆ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡣ㸪ࢫ࣮ࣜࣈእᚄ࡜ྠ୍
ᚄࡢࢯࣜࢵࢻ࡞㍈ࡢᅛ᭷᣺ືᩘ࡜࡯ࡰ୍⮴ࡍࡿ࡜࠸࠺⤖ᯝࡀᚓࡽࢀࡓ㸬
ࡋࡓࡀࡗ࡚㸪ᔐࡵྜ࠸㒊ࢆࡶࡘ㍈ࡢ᭤ࡆ᣺ືࢆ⪃࠼ࡿሙྜ㸪ᔐࡵྜ࠸㒊࡟㐣኱࡞᭤ࡆࣔ
࣮࣓ࣥࢺࡀస⏝ࡋ࡞ࡅࢀࡤࡍ࡭ࡾࡀ⏕ࡌ࡞࠸ࡓࡵ㸪ᔐࡵྜ࠸㒊ࢆࢫ࣮ࣜࣈእᚄ࡜ྠ୍ᚄࡢ
ࢯࣜࢵࢻ࡞㍈࡜ࡋ࡚ᢅ࠼ࡤࡼ࠸ࡇ࡜ࡀ᫂ࡽ࠿࡟࡞ࡗࡓ㸬
59
➨ ❶ ㍈┤ᚄኚ໬㒊ࡢࡡࡌࡾ๛ᛶࡢ㧗⢭ᗘ໬
⥴ゝ
➨ 3 ❶࡛ࡣ㸪㧗⏣ࡽࡢ᪤Ꮡ◊✲ࢆᇶ࡟ࡋ࡚㸪1D FEM ࣔࢹࣝ࡟࠾࠸࡚㸪㍈┤ᚄኚ໬㒊ࡢᑠ
ᚄ㒊࡟ 9d/32 ࡢせ⣲ࢆタࡅ㸪ࡑࡢ㒊ศࡢ᭤ࡆ๛ᛶࢆᙅࡵࡿࡇ࡜࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢ
ᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶࡢపୗࢆ⾲⌧ࡍࡿ᪉ἲࢆ♧ࡋࡓ㸬ᮏ❶࡛ࡣ㸪㍈┤ᚄ
ኚ໬㒊ࡢࡡࡌࡾ๛ᛶࡢ㧗⢭ᗘ໬ࢆ┠ⓗ࡜ࡋ࡚㸪᭤ࡆࡢሙྜ࡜ྠᵝࡢ⪃࠼᪉ࢆ⏝࠸࡚㸪㍈┤
ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢࡡࡌࡾ๛ᛶࡢపୗ࡟ࡘ࠸᳨࡚ウࡍࡿ㸬࣮ࣟࢱ
ࡢࡡࡌࡾ᣺ືࢆᢅ࠺ୖ࡛㸪㍈┤ᚄኚ໬㒊ࡢࡡࡌࡾ๛ᛶపୗ࡟㛵ࡍࡿ◊✲ࡣⴭ⪅ࡢ▱ࡿ㝈ࡾ
ぢᙜࡓࡽ࡞࠸㸬
5.2 ⠇࡛ࡣ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗࢆ㸪࡝ࡢࡼ
࠺࡞ၥ㢟࡜ࡋ࡚ᢅ࠺ࡇ࡜ࡀ࡛ࡁࡿ࠿ࢆ♧ࡍ㸬➃㠃୰ኸ㒊࡟ᅛᐃࡉࢀࡓ๛య㍈࡟ࡼࡗ࡚㟼ⓗ
ࢺࣝࢡࢆཷࡅࡿ㸪እ࿘㠃ࢆᅛᐃࡉࢀࡓ෇ᰕࡢᙎᛶኚᙧࡢ⌮ㄽᘧࢆ㸪ⷧᯈ࡜ཌᯈࡢࡑࢀࡒࢀ
࡟ࡘ࠸࡚♧ࡍ㸬
5.3 ⠇࡛ࡣ㸪ᥦ᱌ࡍࡿ㍈┤ᚄኚ໬㒊ࡢࡡࡌࡾ๛ᛶࢆ㧗⢭ᗘ໬ࡍࡿ᪉ἲࡢ㐺⏝⠊ᅖ࡟ࡘ࠸࡚
᳨ウࡍࡿ㸬3D FEM ࣔࢹࣝ࡟ࡼࡿゎᯒ⤖ᯝࢆ㸪๛య㍈࡟ࡼࡗ࡚➃㠃୰ኸ㒊࡟㟼ⓗࢺࣝࢡࢆཷ
ࡅࡿእ࿘㠃ࢆᅛᐃࡉࢀࡓⷧᯈ࠾ࡼࡧཌᯈࡢᙎᛶኚᙧࡢ⌮ㄽᘧ࡜ẚ㍑ࡋ࡚㸪ᥦ᱌᪉ἲࡢ㐺⏝
⠊ᅖࢆ᫂ࡽ࠿࡟ࡍࡿ㸬
5.4 ⠇࡛ࡣ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗࢆ⪃៖ࡋࡓ
ࣔࢹࣝ໬᪉ἲ࡟ࡘ࠸᳨࡚ウࡍࡿ㸬ࡲࡎ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡜➼౯࡞ࡡࡌࡾ๛ᛶ
࡟ࡘ࠸᳨࡚ウࡍࡿ㸬ḟ࡟㸪ᅇ㌿ࡤࡡࡢ┤ิ⤖ྜ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼ
ࡿࡳ࠿ࡅࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝࢆ᳨ウࡍࡿ㸬ࡑࡋ࡚㸪ᅇ㌿ࡤࡡࡢ୪ิ⤖
ྜ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗࢆ⪃៖ࡋࡓࣔ
ࢹࣝࢆ᳨ウࡍࡿ㸬ࡉࡽ࡟㸪➼౯࡞ࡡࡌࡾ๛ᛶࢆ୚࠼ࡿ㍈ᚄ࡟ࡘ࠸᳨࡚ウࡍࡿ㸬
5.5 ⠇࡛ࡣ㸪ᥦ᱌᪉ἲࡢጇᙜᛶࢆ᳨ドࡍࡿࡓࡵ࡟㸪ࡡࡌࡾ๛ᛶࢆ㧗⢭ᗘ໬ࡋࡓ 3D FEM ࣔ
ࢹࣝࢆཧ↷ࡋ࡚㸪ࡡࡌࡾ࣮ࣔࢻࡢᅛ᭷᣺ືᩘ࡜㟼ⓗࢺࣝࢡ࡟ࡼࡿࡡࡌࡾゅࢆホ౯ࡍࡿ㸬ࡉ
ࡽ࡟㸪ලయ౛࡜ࡋ࡚ᐇᶵ┦ᙜࡢゎᯒࣔࢹࣝ࡟ࡘ࠸࡚㸪ᥦ᱌᪉ἲࢆ㐺⏝ࡋࡓሙྜ࡜㐺⏝ࡋ࡞
࠸ሙྜࡢࡡࡌࡾ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࢆẚ㍑ࡋ࡚㸪࡝ࡢ⛬ᗘࡢᕪ␗ࡀ⏕ࡌࡿ࠿ࢆ♧ࡍ㸬
㟼ⓗࢺࣝࢡ࡟ࡼࡿእ࿘ᅛᐃ෇ᰕࡢᙎᛶኚᙧ
ᅗ 5.1 ࡣ㸪➃㠃୰ኸ㒊࡟ᅛᐃࡉࢀࡓ๛య㍈࡟ࡼࡗ࡚㟼ⓗࢺࣝࢡࢆཷࡅࡿ㸪እ࿘㠃ࢆᅛᐃࡉ
ࢀࡓ෇ᰕࡢᙎᛶኚᙧࡢᵝᏊࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬ࡇࡇ࡛㸪๛య㍈ࡢ༙ᚄࡣ a㸪ᙎᛶ෇ᰕࡢ༙
ᚄࡣ b㸪ᙎᛶ෇ᰕࡢཌࡉࡣ h㸪๛య㍈࡟స⏝ࡍࡿࢺࣝࢡࡣ T㸪๛య㍈ࡢᅇ㌿ゅࡣ ș ࡛࠶ࡿ㸬
ࡇࡢࣔࢹ࡛ࣝࡣ㸪༙ᚄ b㸦┤ᚄ 2b=d0㸧ࡢᙎᛶ෇ᰕࡀ㍈┤ᚄኚ໬㒊ࡢ኱ᚄ㍈࡟┦ᙜࡋ㸪༙ᚄ
a㸦┤ᚄ 2a=d㸧ࡢ๛య㍈ࡀ㍈┤ᚄኚ໬㒊ࡢᑠᚄ㍈࡟┦ᙜࡍࡿ㸬๛య㍈ࡢᅇ㌿ゅ ș ࡀࢺࣝࢡ T
࡟ࡼࡿ㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡛࠶ࡿ㸬࡞࠾㸪ᅗ 5.1 ࡣ MATLAB®ࡢࢧ࣮ࢻࣃ࣮ࢸ࢕
Toolbox ࡛࠶ࡿ SDTools ♫〇ࡢ Structural Dynamics Toolbox ࢆ౑⏝ࡋ࡚సᡂࡋࡓ 3D FEM ࣔࢹ
60
࠶ࡿ㸬
࡛ࣝ࠶
y
ș
F
Fixed
outer surrface
Thicckness: h
b
a
T
x
z
Fiig. 5.1
Defformation of a elastic cylinnder which has
h fixed outer surface duee to applied to
orque
on its end surfface by a rigid
d shaft fixed on it
ᛶ෇ᰕࡢཌࡉ
ࡉ h ࡀⷧ࠸ሙ
ሙྜࡣ㸪2 ḟ
ḟඖࡢᙎᛶኚ
ኚᙧၥ㢟࡜ࡋ࡚ᢅ࠺ࡇ࡜ࡀ
ࡀ࡛ࡁ㸪ࢺࣝ
ࣝࢡ T
ᙎᛶ
࡜๛య
య㍈ࡢᅇ㌿ゅ
ゅ ș ࡢ㛵ಀࡣ
ࡣᘧ㸦5.1㸧࡛
࡛⾲ࡉࢀࡿ㸦୰ཎ, 1977㸧㸬
1
T § 1
¨ 2 2
4SGh © b
a
T
·
¸ ¹
㸦5.1㸧
G㸸ᶓᙎᛶ
ᛶಀᩘ (N/m2)
ࡀ࠶ࡿ㸬Sneeddon,
ᙎᛶ
ᛶ෇ᰕࡢཌࡉ
ࡉ h ࡀཌ࠸ሙ
ሙྜࡣ㸪3 ḟ
ḟඖࡢᙎᛶኚ
ኚᙧၥ㢟࡜ࡋ࡚ᢅ࠺ᚲせࡀ
I. N.ࡣ㸪b>>a ࡢሙྜࡢࢺࣝ
ࡢ
ࣝࢡ T’ࡀᘧ㸦
㸦5.2㸧࡛⾲ࡉ
ࡉࢀ㸪ࢺࣝࢡ
ࢡ T ࡜ T’ࡢẚ
ẚ T/T’ࡀᘧ㸦5.3㸧
ࡢ㏲ḟ
ḟ㏆ఝゎ࡜ࡋ
ࡋ࡚⾲ࡉࢀࡿ
ࡿࡇ࡜ࢆ♧ࡋ
ࡋ࡚࠸ࡿ㸦Sn
neddon, I. N., 1966㸧㸬
Tf
16
Ga 3T 㸦5.2㸧
3
61
T
Tf
1 0.3382 E 3 0.0815E 5 0.1144E 6 0.0125E 7 O E 8 㸦5.3㸧
ȕ㸸㍈┤ᚄẚ㸪 E
ab
d d0 ᮏ◊✲࡛ࡣ㸪ᘧ㸦5.2㸧࡜ᘧ㸦5.3㸧ࡢ O(ȕ8)ࡢ㡯ࢆ┬࠸ࡓᘧ࠿ࡽᚓࡽࢀࡿ㸪ᘧ㸦5.4㸧ࢆࢺ
ࣝࢡ T ࡜๛య㍈ࡢᅇ㌿ゅ ș ࡢ㛵ಀᘧ࡜ࡍࡿ㸬
T
3T
㸦5.4㸧
16Ga 1 0.3382E 0.0815E 5 0.1144E 6 0.0125E 7
3
3
㐺⏝⠊ᅖࡢ᳨ウ
ḟ⠇௨㝆࡟♧ࡍᥦ᱌᪉ἲࡣ㸪ᘧ(5.4)ࡀᡂ❧ࡍࡿ㸪኱ᚄ㒊ࢆཌᯈ࡜ࡋ࡚ᢅ࠺ࡇ࡜ࡀ࡛ࡁࡿሙ
ྜࢆ᝿ᐃࡋ࡚࠸ࡿ㸬ࡑࡇ࡛㸪኱ᚄ㒊ࢆཌᯈ࡜ࡋ࡚ᢅ࠺ࡇ࡜ࡀ࡛ࡁࡿ⠊ᅖࢆ᳨ウࡍࡿࡓࡵ࡟㸪
ᅗ 5.1 ࡟♧ࡋࡓ 3 ḟඖ᭷㝈せ⣲ࣔࢹࣝࢆ⏝࠸࡚㸪㍈┤ᚄẚ ȕ=0.1500㸪0.3175㸪0.5650㸪0.7874
ࡢሙྜ࡟ࡘ࠸࡚㸪๛య㍈࡟స⏝ࡉࡏࡿࢺࣝࢡࢆ T=1000 Nm ࡜ࡋ㸪኱ᚄ㒊ࡢᖜᚄẚ h/d0 ࢆ 0.01
࠿ࡽ 1 ࡲ࡛ኚ໬ࡉࡏࡓሙྜࡢ๛య㍈ࡢᅇ㌿ゅ ș ࢆồࡵࡓ㸬ࡑࡢ⤖ᯝࢆᅗ 5.2 ࡟♧ࡍ㸬ྠᅗ࡟
ࡣ㸪ⷧᯈࡢࢺࣝࢡ T ࡜ᅇ㌿ゅ ș ࡢ㛵ಀᘧ㸦5.1㸧ࡢ್ࢆ㟷Ⰽࡢᐇ⥺࡛♧ࡋ㸪ཌᯈࡢࢺࣝࢡ T
࡜ᅇ㌿ゅ ș ࡢ㛵ಀᘧ㸦5.4㸧ࡢ್ࢆ㉥Ⰽࡢᐇ⥺࡛♧ࡋࡓ㸬
62
Ḙ E
Equation (5.1))
Ḙ E
Equation (5.4))
0.2
0.02
F
Fig. 5.2
Reelationship beetween the rottation angle of
o rigid shaft and the widthh – diameter ratio
on elastic cylinder
c
whicch has fixed oouter surface in case that applied
a
torquee T=1000 Nm
m
ᖜᚄ
ᚄẚ h/d0 ࡀ⣙
⣙ 0.02 ௨ୗࡢ⠊ᅖ࡛ࡣ㸪
㸪ゎᯒ⤖ᯝࡣ
ࡣ㸪࡝ࡢ㍈┤
┤ᚄẚ ȕ ࡟࠾
࠾࠸࡚ࡶⷧᯈ
ᯈࡢࢺ
ࣝࢡ T ࡜ᅇ㌿ゅ ș ࡢ㛵ಀࡀᘧ
ᘧ㸦5.1㸧࡜ Ⰻࡃ୍⮴ࡋ࡚࠸ࡿࡇ࡜ࢆ
ࢆ♧ࡋ࡚࠸ࡿ
ࡿ㸬ࡇࢀࡼࡾ
ࡾ㸪ⷧ
ࡳ࡞ࡏࡿ h/d0 ࡢ⠊ᅖࡣ㍈
㍈┤ᚄẚ ȕ ࡟
࡟ࡼࡽࡎ h d 0 d 0.02 ࡛࠶
࠶ࡿࡇ࡜ࡀศ
ศ࠿ࡿ㸬
ᯈ࡜ࡳ
ᖜᚄ
ᚄẚ h/d0 ࡀ⣙
⣙ 0.2 ௨ୖࡢ
ࡢ⠊ᅖ࡛ࡣ㸪ゎᯒ⤖ᯝࡣ㸪࡝ࡢ㍈┤ᚄ
ᚄẚ ȕ ࡟࠾࠸
࠸࡚ࡶཌᯈࡢ
ࡢࢺࣝ
ࢡ T ࡜ᅇ㌿ゅ ș ࡢ㛵ಀࡀᘧ㸦5.4㸧࡜Ⰻ ࡃ୍⮴ࡋ࡚࠸
࠸ࡿࡇ࡜ࢆ♧
♧ࡋ࡚࠸ࡿ㸬 ࡇࢀࡼࡾ㸪ཌᯈ
࡜ࡳ࡞
࡞ࡏࡿ h/d0 ࡢ⠊ᅖࡣ㍈┤
ࡢ
┤ᚄẚ ȕ ࡟ࡼ
ࡼࡽࡎ h d 0 t 0.2 ࡛࠶ࡿࡇ࡜ࡀศ࠿ࡿ
ࡿ㸬ࡲࡓ㸪ᘧ㸦5.4㸧
࠿ࡽࡶ᫂ࡽ࠿࡛࠶
࠶ࡿࡀ㸪ཌᯈ
ᯈ࡜ࡳ࡞ࡏࡿ
ࡿ⠊ᅖ࡛ࡣ㸪୍ᐃࡢࢺࣝࢡ࡟ᑐࡍࡿ๛
๛య㍈ࡢᅇ㌿
㌿ゅ ș
୍ᐃ࡜࡞ࡿ㸬
ࡣ h/d0 ࡟ࡼࡽࡎ୍
ᐇ㝿
㝿ࡢᅇ㌿㍈࡛
࡛ࡣ㸪኱ᚄ㒊
㒊ࡀⷧᯈ࡜ࡳ
ࡳ࡞ࡏࡿሙྜ
ྜࡣࡡࡌࡾ๛
๛ᛶపୗࡣ↓
↓ど࡛ࡁࡿࡀ
ࡀ㸪኱
ᚄ㒊ࡀ
ࡀཌᯈ࡜ࡳ࡞
࡞ࡏࡿሙྜࡣ
ࡣࡡࡌࡾ๛ᛶ
ᛶపୗࡀ㉳ࡇ
ࡇࡿ㸬ࡼࡗ࡚
࡚㸪኱ᚄ㒊ࡀ
ࡀཌᯈ࡜ࡳ࡞
࡞ࡏࡿ
ሙྜ࡟
࡟ࡘ࠸࡚ࡢࡳ
ࡳ㸪㍈┤ᚄኚ
ኚ໬㒊᩿㠃ࡢ
ࡢᙎᛶኚᙧ࡟
࡟ࡼࡿࡡࡌࡾ๛ᛶపୗࢆ
ࢆ⪃៖ࡍࢀࡤ
ࡤࡼ࠸㸬
ࡑࡢ⠊
⠊ᅖࡣ㸪㍈┤
┤ᚄẚ ȕ ࡟ࡼ
ࡼࡽࡎ㸪኱ᚄ㒊
㒊ࢆཌᯈ࡜ࡳ
ࡳ࡞ࡍࡇ࡜ࡀ
ࡀ࡛ࡁࡿ h d 0 t 0.2 ࡛࠶
࠶ࡿ㸬
63
㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧ࡟ࡼ
ࡼࡿࡳ࠿ࡅࡢ
ࡢ㍈ࡢࡡࡌࡾ
ࡾ๛ᛶపୗࢆ
ࢆ⪃៖ࡋࡓࣔ
ࣔࢹࣝ
ἲ
໬᪉ἲ
㍈┤ᚄኚ
ኚ໬㒊᩿㠃ࡢ
ࡢᙎᛶኚᙧࡢ
ࡢ➼౯ࡡࡌࡾ๛ᛶ
๛య
య㍈࡟స⏝ࡍ
ࡍࡿࢺࣝࢡ T ࡟ࡼࡗ࡚⏕
⏕ࡌࡿ๛య㍈
㍈ࡢᅇ㌿ゅ ș㸦ᅗ 5.1 ཧ↷
↷㸧ࢆ㸪∦ഃ
ഃࡢ➃
㠃ࡀᅛᐃࡉࢀࡓ୰
୰ᐇ෇ᰕࡢࡡ
ࡡࡌࡾゅ࡜➼
➼౯࡜࠾࠸࡚
࡚㸪㍈┤ᚄኚ
ኚ໬㒊᩿㠃ࡢ
ࡢᙎᛶኚᙧࡢ
ࡢ➼౯
ࡵࡿ㸬
ࡡࡌࡾ๛ᛶࢆồࡵ
ᚄ㒊ࡀཌᯈࡢ
ࡢሙྜࡣ㸪๛
๛య㍈࡟స⏝
⏝ࡍࡿࢺࣝࢡ
ࢡ T ࡟ࡼࡗ࡚
࡚⏕ࡌࡿ๛య
య㍈ࡢᅇ㌿ゅ
ゅșࡢ
኱ᚄ
㛵ಀࡣ
ࡣᘧ㸦5.4㸧࡛୚࠼ࡽࢀࡿ㸬
ᅗ 5.3 ࡟♧ࡉࢀ
ࢀࡿ㸪㛗ࡉ l㸪ᶓᙎᛶಀᩘ
ᩘ G㸪᩿㠃 2 ḟᴟ࣮࣓ࣔ
࣓ࣥࢺ Ip ࡢ㸪 ୍᪉ࡢ➃㠃
㠃ࡀᅛ
ࢀࡓ୰ᐇ෇ᰕ
ᰕࡀ㸪ࡶ࠺୍
୍᪉ࡢ➃㠃࡟
࡟ࢺࣝࢡ T ࢆཷࡅࡓ࡜ࡁ
ࢆ
ࡁࡢඛ➃ࡢࡡ
ࡡࡌࡾゅ ș ࡣ㸪ᘧ
ࡣ
ᐃࡉࢀ
㸦5.55㸧࡛୚࠼ࡽࢀࡿ㸦୰ἑ௚
௚, 1973b㸧㸬
T
Fixed
d
G , Ip
ș
l
Fig. 5.3 Torsional aangle of cantillever beam du
ue to torque
T
Tl
GI p
332Tl
GSd 4
㸦5.5㸧
ᘧ㸦5.4㸧࡜ᘧ㸦5.5㸧࠿ࡽ㸪㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧࡢ➼
➼౯ࡡࡌࡾ๛
๛ᛶ GIpd ࢆ⾲
⾲ࡍᘧ
㸦5.66㸧ࡀᚓࡽࢀࡿ㸬
GI pd
16
Gaa 3 1 0.3382E 3 0.0815E 5 0.1144E 6 0.0125E 7 ˜ l
3
㸦5.6㸧
ᅇ㌿ࡤࡡ
ࡡࡢ┤ิ⤖ྜ
ྜ࡟ࡼࡗ࡚㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧ࡟ࡼࡿ
ࡿࡳ࠿ࡅࡢ㍈
㍈ࡢࡡ
ࡌࡾ๛
๛ᛶపୗࢆ⪃
⪃៖ࡋࡓࣔࢹ
ࢹࣝ
1D
D FEM ࣔࢹࣝ
ࣝ࡟࠾࠸࡚㸪㍈┤ᚄኚ໬ 㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧ࡟ࡼࡿ
ࡿࡳ࠿ࡅࡢ㍈
㍈ࡢࡡࡌࡾ๛
๛ᛶప
64
⪃៖ࡋࡓࣔࢹ
ࢹࣝࢆᅗ 5.4
4 ࡟♧ࡍ㸬
ୗࢆ⪃
A
h
ȭd 0
Ip1
Ip”=Ip1’+Ip2’
1
F
Fig. 5.4
Ip2
3
2
element 1
l2
ȭd
element 2
A
1D beam model for torsion inn consideratio
on for elasticiity of steppedd cross-sectio
on
3.33.2 ⠇࡛♧ࡋ
ࡋࡓ᭤ࡆࡢሙྜ࡜ྠᵝ࡟㸪
㸪3 ࡘࡢ⠇Ⅼ
Ⅼ࡛ 2 ࡘࡢ㍈
㍈せ⣲ࢆᐃ⩏
⩏ࡋ㸪኱ᚄ㍈
㍈ྑ➃
ࡢ⠇Ⅼ
Ⅼ 2 ࡢࡡࡌࡾ๛ᛶ GIp1’࡜ᑠᚄ㍈ࡢᕥ
ᕥ➃ࡢ⠇Ⅼ 2 ࡢࡡࡌࡾ๛
๛ᛶ GIp2’ࢆࡑ
ࡑࢀࡒࢀᘧ㸦5.7㸧
࡜ᘧ㸦5.8㸧ࡢ┤ิ⤖ྜ࡟ࡼࡗ࡚ồࡵࡿ㸬
GI cp1
G
I p1 I pd
I p1 I pd
㸦5.7㸧
GI cp 2
G
I p 2 I pd
I p 2 I pd
㸦5.8㸧
࡟㸪኱ᚄ㍈࡜
࡜ᑠᚄ㍈ࡢ⤖
⤖ྜ㒊㸦⠇Ⅼ
Ⅼ 2㸧ࡢࡡࡌࡾ๛ᛶࢆ㸪㏻
㏻ᖖࡢ㔜ࡡྜ
ྜࢃࡏ㸪ࡍ࡞
࡞ࢃࡕ
ḟ࡟
୪ิ⤖
⤖ྜ࡟ࡼࡗ࡚
࡚ồࡵࡿ㸬
GI p
s
GI cp1 GI cp 2
§ I p1 I pd
I p 2 I pd
G¨
¨ I p1 I pd I p 2 I pd
©
·
¸
¸
¹
㸦5.9㸧
ᘧ㸦5.9㸧࡟࠾࠸࡚㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧࢆ⾲
⾲ࡍ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥ
ḟ
ࣥࢺ Ipd ࡣᘧ㸦5.6㸧
࡛୚࠼ࡽࢀࡿ㸬
ࡑࡢ
ࡢḟ࡟㸪᭤ࡆ
ࡆࡢሙྜ࡜ྠ
ྠᵝ࡟ࡋ࡚㸪ࡡࡌࡾ๛ᛶ
ᛶࢆᙅࡵࡿᑠᚄ㍈せ⣲ࡢ㛗
㛗ࡉ l ࡢጇᙜ
ᙜ࡞್
ࢆ᳨ウ
ウࡍࡿ㸬ᘧ㸦5.4㸧࡜ᘧ㸦5.5㸧ࡼࡾ 㸪㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧ࡜➼
➼౯࡞ࡡࡌࡾ
ࡾ๛ᛶ
ࢆ୚࠼ࡿᑠᚄ㍈ࡢ
ࡢ㛗ࡉ l ࢆ⾲
⾲ࡍᘧ㸦5.100㸧ࡢ᮲௳ࡀᚓ
ᚓࡽࢀࡿ㸬
65
l
3Sa
32 1 0.3382E 0.0815E 5 0.1144E 6 0.0125E 7
㸦5.10㸧
3Sd
64 1 0.3382E 3 0.0815E 5 0.1144E 6 0.0125E 7
3
ᘧ㸦5.10㸧࡛୚࠼ࡽࢀࡿ㛗ࡉ l ࢆᣢࡘ┤ᚄ d ࡢᑠᚄ㍈ࡢせ⣲ࡢࡡࡌࡾ๛ᛶ GIp2 ࡀ㍈┤ᚄኚ
໬㒊᩿㠃ࡢࡡࡌࡾ๛ᛶ GIpd ࡜➼౯࡛࠶ࡿࡓࡵ㸪
GI pd
GI p 2
㸦5.11㸧
࡜࠾ࡃࡇ࡜ࡀ࡛ࡁ㸪᩿㠃 A-A ࡟࠾ࡅࡿࡡࡌࡾ๛ᛶ GIpA ࢆ⾲ࡍᘧ㸦5.9㸧ࡣ㸪ᘧ㸦5.12㸧࡜࡞
ࡿ㸬
GI pA
GI p
s
§ I p1I p 2
§ I I
I p2I p2 ·
I ·
¸ G¨ p1 p 2 p 2 ¸ 㸦5.12㸧
G¨
¨ I p1 I p 2 I p 2 I p 2 ¸
¨ I p1 I p 2
2 ¸¹
©
¹
©
ᘧ㸦5.12㸧࡟ࡼࡗ࡚㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗࢆ
⪃៖ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࡀ㸪኱ᚄ㍈せ⣲㸪ᑠᚄ㍈せ⣲㸪ඹ࡟㍈せ⣲ࡢ୧➃࡛ࡡࡌࡾ๛ᛶࡀ␗
࡞ࡿࡓࡵ㸦GIp1GIp1’㸪GIp2’GIp2㸧㸪୍⯡ⓗ࡞ 1D FEM ࣔࢹࣝࡀ㐺⏝ࡉࢀࡓࡡࡌࡾ᣺ືゎᯒ
⎔ቃ࡟ࡇࡢ᪉ἲࢆ㐺⏝ࡍࡿࡇ࡜ࡣᐜ࡛᫆ࡣ࡞࠸㸬
ᅇ㌿ࡤࡡࡢ୪ิ⤖ྜ࡟ࡼࡗ࡚㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢࡡ
ࡌࡾ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝ
୍⯡ⓗ࡞ 1D FEM ࣔࢹ࡛ࣝ⏝࠸ࡽࢀ࡚࠸ࡿࡡࡌࡾࡤࡡࡢ୪ิ⤖ྜ࡟ࡼࡗ࡚㸪㍈┤ᚄኚ໬
㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡡࡌࡾ๛ᛶపୗࢆ⪃៖ࡍࡿࡇ࡜ࡀ࡛ࡁࡿࣔࢹࣝ໬᪉ἲࢆᥦ᱌ࡍࡿ㸬
ᅗ 5.5 ࡣ㸪ࡇࡇ࡛ᥦ᱌ࡍࡿ㍈┤ᚄኚ໬㒊ࡢࣔࢹࣝ໬᪉ἲࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬
66
l
A
h
l2’= l2-l
ȭd 0
Ip2
1
element 1
l
3
2
A
4
element 2
3Sd
64 1 0.33382E 0.08
815E 5 0.11444E 6 0.012
25E 7
3
F
Fig. 5.5
ȭd
Ipeq
Ip1
1D beam model for torsion inn consideratio
on for elasticiity of steppedd cross-sectio
on
using
u
the methhod proposed
d by the autho
or
ࡲࡎ
ࡎ㸪ᘧ㸦5.10㸧࡛୚࠼ࡽ
ࡽࢀࡿ㛗ࡉࡢ
ࡢ㍈せ⣲ࢆ㍈
㍈┤ᚄኚ໬㒊ࡢᑠᚄ㒊࡟タ
タࡅࡿ㸬
ḟ࡟
࡟㸪ᘧ㸦5.12㸧࡛♧ࡉࢀ
ࢀࡿ᩿㠃 A-A
A ࡟࠾ࡅࡿࡡ
ࡡࡌࡾ๛ᛶ GI
G pA ࢆ㸪୍⯡
⯡ⓗ࡞ 1D FE
EM ࣔ
ࢹ࡛ࣝ
࡛⏝࠸ࡽࢀࡿ
ࡿ㔜ࡡྜࢃࡏ
ࡏ㸪ࡍ࡞ࢃࡕ
ࡕ୪ิ⤖ྜ࡛
࡛⾲ࡍࡇ࡜ࢆ
ࢆ⪃࠼ࡿ㸬ᅗ 5.5 ࡟࠾࠸࡚
࡚㸪ᘧ
㸦5.10㸧࡛୚࠼ࡽ
ࡽࢀࡿ㛗ࡉࡢ
ࡢᑠᚄ㍈せ⣲
⣲ࡢ➼౯ࡡࡌ
ࡌࡾ๛ᛶࢆ GI
G peq ࡜࠾࠸࡚
࡚⠇Ⅼ 2 ࡜⠇
⠇Ⅼ 3
࡛ྛࠎ
ࠎࡢࡡࡌࡾ๛
๛ᛶࡀ୪ิ⤖
⤖ྜࡉࢀ㸪⠇
⠇Ⅼ 2㸪3 ࡢࡡࡌࡾ๛ᛶ
ࡢ
ᛶࡀ┤ิ⤖ྜ
ྜࡉࢀࡓࡶࡢ
ࡢࡀᘧ
㸦5.13㸧࡛࠶ࡾ㸪ࡇࢀࡣᘧ㸦5.12㸧࡜➼౯
౯࡛࠶ࡿ㸬
GI pA
G
I
I
p1
p1
I
I peq ˜ I p 2 I peq
I peq
p2
㸦5.13㸧
I peq
ࡼࡗ
ࡗ࡛㸪ᘧ㸦5.113㸧࡜ᘧ㸦5.1
12㸧࠿ࡽ㸪ࡡ ࡌࡾ๛ᛶࡢ➼
➼౯᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ
ḟ
ࢺ Ipeq ࡣᘧ㸦5.14
5 㸧
࡜࡞ࡿ㸬
2
I peq
4
2
3
2
I p1 I p 2 I p1 I p1 7 I p1 I p 2 6 I p1 I p 2 2 I p 2
2 I p1 I p 2
4
㸦5.14㸧
ᅗ 5.5 ࡢ⠇Ⅼ 2 ࡜⠇Ⅼ 3 ࡛ᐃ⩏ࡉࢀࡿ
࡛
ࡿ㸪ᘧ㸦5.10㸧࡛୚࠼ࡽࢀ
㸧
ࢀࡿ㛗ࡉࡢ㍈
㍈せ⣲ࢆ㍈┤
┤ᚄኚ
ࡢᑠᚄഃ࡟タ
タࡅ࡚㸪ࡑࡢ
ࡢ㒊ศࡢࡡࡌ
ࡌࡾ๛ᛶࢆ GI
G peq ࡜࠾ࡃࡇ
ࡇ࡜࡛㸪᩿㠃
㠃 A-A ࡢᙎᛶ
ᛶኚᙧ
໬㒊ࡢ
ࢆ⪃៖
៖ࡋࡓࡡࡌࡾ๛ᛶࢆ⾲⌧
⌧ࡍࡿࡇ࡜ࡀ
ࡀ࡛ࡁࡿ㸬ࡓࡔࡋ㸪ᘧ㸦5.1
14㸧ࡀᡂ❧ࡍ
ࡍࡿࡢࡣ㸪ᘧ㸦5.4㸧
67
❧ࡍࡿ࠾ࡼࡑ
ࡑ h/ d0 • 0.2 ࡢ⠊ᅖ࡟࠾
࠾࠸࡚࡛࠶ࡿ㸬
ࡀᡂ❧
๛ᛶࡢ➼౯㍈
㍈ᚄࡢ᳨ウ
ࡡࡌࡾ๛
ᮏ⠇
⠇࡛ࡣ㸪㍈┤
┤ᚄኚ໬㒊ࡢ
ࡢࡳ࠿ࡅࡢ㍈
㍈ࡢࡡࡌࡾ๛
๛ᛶపୗࢆᢕ
ᢕᥱࡋ᫆ࡃࡍ
ࡍࡿࡓࡵ࡟㸪➼౯
࡞ࡡࡌࡾ๛ᛶࢆ୚
୚࠼ࡿᑠᚄ㒊
㒊ࡢ㍈ᚄ dpeqq ࢆᐃ⩏ࡋ㸪dpeq ࡀ㍈┤ᚄ
ᚄẚ ȕ (=d/d00)࡟ᑐࡋ࡚࡝
࡝ࡢࡼ
ኚ໬ࡍࡿ࠿ࢆ
ࢆ᳨ウࡍࡿ㸬
࠺࡟ኚ
➼౯
౯࡞ࡡࡌࡾ๛
๛ᛶࢆ୚࠼ࡿ
ࡿᑠᚄ㒊ࡢ㍈
㍈ᚄࢆ dpeq ࡜ࡋ࡚㸪ᘧ㸦
࡜
5.14㸧࡛♧ࡉ
ࡉࢀࡿࡡࡌࡾ
ࡾ๛ᛶ
ࡢ➼౯
౯᩿㠃 2 ḟᴟ
ᴟ࣮࣓ࣔࣥࢺ Ipeq ࢆ୰ᐇ
ᐇ෇ᙧ᩿㠃㍈
㍈ࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥ
ḟ
ࣥࢺࡢᙧ࡛⾲ࡍ࡜㸪
ᘧ㸦55.15㸧࡜࡞ࡿ
ࡿ㸬
I peq
S
4
32
d peeq 㸦5.15㸧
ᘧ㸦55.15㸧ࡼࡾ㸪➼౯࡞ࡡࡌ
ࡌࡾ๛ᛶࢆ୚
୚࠼ࡿᑠᚄ㒊
㒊ࡢ㍈ᚄ dpeq ࡣᘧ㸦5.16㸧 ࡛⾲ࡉࢀࡿ
ࡿ㸬
d peq
4
32
S
I peq
p
㸦5.16㸧
ᘧ㸦5.14㸧࡟ Ip1=ʌd04/32㸪Ip2=ʌd4/32 ࢆ௦
௦ධࡋ࡚ Ipeq ࢆ d0 ࡜ d ࡢ㛵
㛵ᩘ࡛⾲ࡋ㸪㍈
㍈┤ᚄẚ ȕ (=
( d/d0)
౯㍈ᚄẚ dpeeq/d ࡢ㛵ಀࢆ
ࢆồࡵࡿ࡜㸪 ᭤ࡆࡢሙྜ
ྜࡢ㍈┤ᚄẚ ȕ ࡜➼౯㍈
㍈ᚄẚ deq/d ࡢ㛵ಀ
ࡢ
࡜➼౯
㸦ᅗ 3.9 ཧ↷㸧࡜
࡜ྠ୍ࡢᅗ 5.6 ࡢ᭤⥺ࡀᚓ
ᚓࡽࢀࡿ㸬ྠ
ྠᅗࡼࡾ㸪ᘧ
ᘧ㸦5.4㸧ࡀᡂ
ᡂࡾ❧ࡘ h/d0 • 0.2
ᅖ࡛ࡣ㸪኱ᚄ
ᚄ㍈┤ᚄ d0 ࡀᑠᚄ㍈┤ᚄ
ᚄ d ࡟ᑐࡋ࡚࠸ࡃࡽ኱ࡁ
ࡁࡃ࡞ࡗ࡚ࡶ
ࡶ㸪➼౯࡞ࡡ
ࡡࡌࡾ
ࡢ⠊ᅖ
๛ᛶ GIpeq ࡣᑠᚄ
ᚄ㍈ࡢࡡࡌࡾ๛ᛶ GIp2 ࡢ 0.84094=0.4
4979 ಸࢆୗᅇ
ᅇࡽ࡞࠸࡜ゝ
ゝ࠼ࡿ㸬
Fig
g. 5.6
Relattionship betw
ween ȕ and dpeeq/d
68
௨ୖ࡟ࡼࡾ㸪ᘧ㸦5.10㸧࡛୚࠼ࡽࢀࡿ㛗ࡉࡢᑠᚄ㍈せ⣲ࡢࡡࡌࡾ๛ᛶࢆ GIpeq ࡜ࡍࡿ㸪࠶
ࡿ࠸ࡣ㸪ᘧ㸦5.10㸧࡛୚࠼ࡽࢀࡿ㛗ࡉࡢᑠᚄ㍈せ⣲ࡢ┤ᚄࢆᘧ㸦5.16㸧ࡢ➼౯࡞ࡡࡌࡾ๛ᛶ
ࢆ୚࠼ࡿᑠᚄ㒊ࡢ㍈ᚄ dpeq ࡜ࡍࢀࡤ㸪୍⯡ⓗ࡞୍ḟඖᱱࣔࢹࣝࡀ㐺⏝ࡉࢀࡓ㍈᣺ືゎᯒࢯࣇ
ࢺ࢙࢘࢔࡛㸪㍈┤ᚄኚ໬㒊ࡢᙎᛶኚᙧࢆ⪃៖ࡋࡓ㍈᣺ື㸦ࡡࡌࡾ᣺ື㸧ࡢゎᯒࢆ⡆౽࡟ᐇ
⌧ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
ᥦ᱌᪉ἲࡢጇᙜᛶࡢ᳨ド
ḟඖ᭷㝈せ⣲ࣔࢹࣝࡢࡡࡌࡾ๛ᛶࡢ⿵ṇ
᳨ド࡟ඛ❧ࡕ㸪᭤ࡆࡢሙྜ࡜ྠ୍ࡢ 3D FEM ࣔࢹࣝ㸦ᅗ 3.10 ཧ↷㸧ࡢࡡࡌࡾ᣺ື≉ᛶࢆ
3.4.1 ⠇࡜ྠᵝࡢ᪉ἲ࡛㧗⢭ᗘ໬ࡍࡿ㸬ᮦᩱ≉ᛶࡣ㸪᭤ࡆࡢሙྜ࡜ྠ୍ࡢᶓᙎᛶಀᩘ G=
8.0335×1010 N/m2㸪࣏࢔ࢯࣥẚ Ȟ=0.3008㸪ᐦᗘ ȡ=7846 kg/m3 ࡜ࡍࡿ㸬╔┠ࡍࡿ࣮ࣔࢻḟᩘ࡟
ࡘ࠸࡚ࡣ㸪୍⯡࡟㸪ࡡࡌࡾ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡣ᭤ࡆ࡜ẚ࡭࡚㧗࠸ࡓࡵ㸪ࡇࡇ࡛ࡣ 3 ḟࡲ
࡛࡜ࡍࡿ㸬ࡡࡌࡾ᣺ື≉ᛶࡀ㧗⢭ᗘ࡛࠶ࡿ୍ᵝ᩿㠃ࡢ 1D FEM ࣔࢹࣝ࡟ࡘ࠸࡚ࡶ᭤ࡆࡢሙ
ྜ࡜ྠ୍ࡢࣔࢹࣝ㸦ᅗ 3.11 ཧ↷㸧࡜ࡍࡿ㸬
୍ᵝ࡞୰ᐇ෇ᙧ᩿㠃㍈ࡢ 1D FEM ࣔࢹࣝࡢࡡࡌࡾ 1 ḟ㹼3 ḟࡢᅛ᭷᣺ືᩘࢆཧ↷ࡋ࡚ 3D
FEM ࣔࢹࣝࡢᶓᙎᛶಀᩘࢆ⿵ṇࡍࡿ㸬ᶓᙎᛶಀᩘࡢ⿵ṇ࡟ࡣ㸪ᘧ㸦5.17㸧ࡢᅛ᭷ゅ᣺ືᩘ Ȧn
࡜ࡡࡌࡾ๛ᛶ GIp ࡢ㛵ಀ࠿ࡽᚓࡽࢀࡿᘧ㸦5.18㸧ࢆ⏝࠸ࡿ㸬
2
Zn v GI p 㸦5.17㸧
2
§ 1 ·
Gc ¨
¸ ˜ G 㸦5.18㸧
©1 H ¹
G 㸸⿵ṇ๓ࡢᶓᙎᛶಀᩘ (N/m2)
G’㸸⿵ṇᚋࡢᶓᙎᛶಀᩘ (N/m2)
H 㸸ࡡࡌࡾ 1 ḟ㹼3 ḟࡢᅛ᭷ゅ᣺ືᩘࡢᖹᆒㄗᕪ
⾲ 5.1 ࡣ㸪ᅗ 3.11 ࡢ 1D FEM ࣔࢹࣝ㸦ཧ↷ࣔࢹࣝ㸧࡜ᅗ 3.10 ࡢ 3D FEM ࣔࢹࣝࡢࡡࡌࡾ
1 ḟ㹼3 ḟࡢᅛ᭷᣺ືᩘࡢẚ㍑ࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬3D Model 0 ࡀᶓᙎᛶಀᩘࢆ⿵ṇࡍࡿ๓
ࡢ G=8.0335×1010 N/m2 ࡜ࡋࡓࣔࢹ࡛ࣝ࠶ࡾ㸪3D Model 1 ࡀᶓᙎᛶಀᩘࢆᘧ㸦5.18㸧࡟ࡼࡗ࡚
⿵ṇࡋࡓ G’=8.0726×1010 N/m2 ࡜ࡋࡓࣔࢹ࡛ࣝ࠶ࡿ㸬1D FEM ࣔࢹࣝ࡟ᑐࡍࡿ㸪ᶓᙎᛶಀᩘ
ࢆ⿵ṇࡍࡿ๓ࡢ 3D FEM ࣔࢹࣝࡢᅛ᭷᣺ືᩘࡢㄗᕪࡢ⤯ᑐ್ࡣ᭱኱࡛ 0.48%࡛࠶ࡿࡢ࡟ᑐࡋ
࡚㸪ᶓᙎᛶಀᩘࢆ⿵ṇࡋࡓᚋࡢ 3D FEM ࣔࢹࣝࡢᅛ᭷᣺ືᩘࡢㄗᕪࡢ⤯ᑐ್ࡣ᭱኱࡛ 0.24%
࡜ 1/2 ࡟పῶࡋ࡚࠸ࡿ㸬
69
nal modes of a shaft
Table 5.1 Compaarison of natuural frequenciies for torsion
with uniforrm cross-secti
tion between 1D model and 3D model
Unit: Hz
Mode
1D Mod
del
3D Model 0 (Error)
(
3D Model 1 (Error)
1
4573.8
8
4571.4
(-0.05%)
4582.5
(0.19%)
2
9161.5
5
9143.7
(-0.19%)
9165.9
(0.05%)
3
13785
5
13718
(-0.48%)
13752
(-0.24%)
ࡡࡌࡾࣔ
࣮ࣔࢻࡢᅛ᭷
᭷᣺ືᩘ
ࡇࡇ࡛ࡣ㸪3.4.2
3
⠇ࡢ㍈ B ࢆྲྀࡾୖ ࡆࡿ㸦ᅗ 5.7
7 ཧ↷㸧㸬
㍈ B ࡢᑍἲࢆ௨ୗ࡟෌ᥖ
ᥖࡍࡿ㸬
࣭S
Shaft B㸹 ඲㛗
඲ 350mm㸪ᑠᚄ㒊እᚄ
ᚄ ȭ20mm㸪
኱
኱ᚄ㒊እᚄ
ȭ25.4mm㸦ȕ =0.7874㸧࡜
࡜ ȭ35.4mm
m㸦ȕ=0.5650㸧
኱
኱ᚄ㒊ࡢ఩⨨
⨨㸸ᕥ➃࠿ࡽ
ࡽ 57.3mm㹼82.7mm㸦h/d0=1㸧㸪
162.3mm
m㹼187.7mm㸦h/d0=0.717
75㸧
Fig. 5.7
3D
D FEM modeel of Shaft B
ᅗ 5.7 ࡢ 3D FE
EM ࣔࢹࣝࡢ
ࡢ⮬⏤ᨭᣢ≧
≧ែࡢᅛ᭷್
್ゎᯒ⤖ᯝ㸦ࡡࡌࡾ 1 ḟ㹼
㹼3 ḟࡢᅛ᭷
᭷᣺ື
ࢆཧ↷ࡋ࡚㸪ᅗ 5.8 ࡢ 1D
D FEM ࣔࢹ ࣝ࡟ᥦ᱌᪉ἲ
ἲࢆ㐺⏝ࡋࡓ
ࡓሙྜࡢᅛ᭷
᭷᣺ືᩘࢆホ
ホ౯ࡋ
ᩘ㸧ࢆ
10
ࡓ㸬33D FEM ࣔࢹ
ࢹࣝࡢᮦᩱ≉
≉ᛶࡣᶓᙎᛶ
ᛶಀᩘ G’= 8.0726×10
8
N/m2㸦⿵ṇ
ṇᚋ㸧㸪࣏࢔ࢯ
ࢯࣥẚ
Ȟ=0.33008㸪ᐦᗘ ȡ=7846 kg/m3 ࡜ࡋ㸪1D F
FEM ࣔࢹࣝ
ࣝࡢᮦᩱ≉ᛶࡣᶓᙎᛶಀᩘ
ᩘ G= 8.0335
5×1010
N/m2㸪࣏࢔ࢯࣥẚ
ẚ Ȟ=0.3008㸪ᐦᗘ ȡ=78446 kg/m3 ࡜ࡋࡓ㸬࡞࠾㸪ᅗ 5.8 ࡢ 11D FEM ࣔࢹ
ࢹ࡛ࣝ
ࡣᕥ➃
➃࠿ࡽ 5㸪10㸪17㸪22 ␒┠ࡢせ⣲㛗
␒
㛗ࢆᘧ㸦5.10
0㸧࡛୚࠼ࡽࢀࡿ㛗ࡉ࡜ ࡋ࡚࠸ࡿ㸬
70
Fig. 5.8
1D
D FEM modeel of Shaft B
ᅗ 5.9 ࡣ㍈ B ࡢࡡࡌࡾ
ࡢ
1 ḟ㹼
ḟ 3 ḟࡢᅛ
ᅛ᭷࣮ࣔࢻࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬
1
1st
2nd
3rd
Deflection
0.5
0
-0.5
-1
0
0.05
0.1
0.15
0.22
0.25
0
0.3
0.35
P
Position of nod
des (m)
Fig. 5.9
Eigennmodes for to
orsion of Shaaft B
⾲ 5.2 ࡣ㸪ᅗ 5.7
5 ࡢ㍈┤ᚄኚ
ኚ໬㒊ࢆ᭷ࡍ
ࡍࡿ㍈ B ࡢ 3D FEM ࣔࢹ
ࢹࣝ㸦ཧ↷ࣔ
ࣔࢹࣝ㸧࡜㸪ᅗ 5.8
EM ࣔࢹࣝ࡟
࡟ࡘ࠸࡚ࡡࡌ
ࡌࡾ 1 ḟ㹼3 ḟࡢᅛ᭷᣺ື
ḟ
ືᩘࡢẚ㍑ࢆ
ࢆ♧ࡋ
ࡢ㍈ B ࡢୗグࡢ 2 ࡘࡢ 1D FE
ࡢ࡛࠶ࡿ㸬
ࡓࡶࡢ
࣭11D Model 1㸸㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧࢆ⪃
⪃៖ࡋ࡞࠸ࣔࢹࣝ
࣭11D Model 3㸸ᥦ᱌᪉ἲ࡟
࡟ࡼࡗ࡚㍈┤
┤ᚄኚ໬㒊᩿
᩿㠃ࡢᙎᛶኚ
ኚᙧࢆ⪃៖ࡋ ࡓࣔࢹࣝ
71
Taable 5.2 Co
omparison of nnatural frequ
uencies for torrsion of Shaft
ft B
Unit:: Hz
M
Mode
3D Mo
odel
1D M
Model 1 (Erro
or)
1D Model
M
3 (Errorr)
1
4591.0
46500.7
(1.30%)
4598.1
(0.16%
%)
2
6538.3
65933.0
(0.84%)
6543.6
(0.08%
%)
3
1478
80
150330
(1.70%)
1488
80
(0.68%
%)
㍈┤
┤ᚄኚ໬㒊᩿
᩿㠃ࡢᙎᛶኚ
ኚᙧࢆ⪃៖ࡋ
ࡋ࡞࠸ 1D Mo
odel 1 ࡛ࡣ᭱
᭱኱࡛ 1.70%
%ࡢㄗᕪࡀ࠶ࡿࡀ㸪
ᥦ᱌᪉
᪉ἲ࡟ࡼࡗ࡚
࡚㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧࢆ⪃
⪃៖ࡋࡓ 1D Model 3 ࡢㄗ
ㄗᕪࡣ 1D Model
M
1
ࡢ 1/22 ᮍ‶ࡢ 0.68%௨ୗ࡛࠶
࠶ࡿ㸬
௨ୖ
ୖ㸪ᅗ 5.8 ࡢ㍈┤ᚄኚ໬
ࡢ
໬㒊ࢆ᭷ࡍࡿ
ࡿ㍈ࡢ 1D FEM ࣔࢹࣝ࡟ࡘ࠸࡚㸪ᅗ 5.7 ࡢ 3D FE
EM ࣔ
ࢹࣝࢆཧ↷ࡋ࡚㸪
㸪㍈┤ᚄኚ໬
໬㒊᩿㠃ࡢᙎ
ᙎᛶኚᙧࢆ⪃
⪃៖ࡋ࡞࠸ሙ
ሙྜ࡜ᥦ᱌᪉
᪉ἲ࡛⪃៖ࡋ
ࡋࡓሙ
ࡘ࠸࡚ࡡࡌࡾ 1 ḟ㹼3 ḟࡢᅛ᭷᣺ື
ḟ
ືᩘࢆẚ㍑ࡋ
ࡋ㸪ᥦ᱌᪉ἲࡢጇᙜᛶࢆ☜
☜ㄆࡋࡓ㸬
ྜ࡟ࡘ
㟼ⓗࢺࣝ
ࣝࢡ࡟ࡼࡿࡡ
ࡡࡌࡾゅࡢホ
ホ౯
ࡇࡇ࡛ࡣ㸪ᅗ 5.8
5 ࡢ㍈ B ࡢ୰ኸࡢ኱ᚄ
ࡢ
ᚄ㒊࡜ྑഃࡢ
ࡢᑠᚄ㒊࠿ࡽ࡞ࡿ㍈㸪ࡍ࡞
࡞ࢃࡕᅗ 5.8
8 ࡢ㍈
B ࡢ⠇
⠇Ⅼ 18 ࠿ࡽ
ࡽ⠇Ⅼ 35 ࡲ࡛
࡛ࡢ㍈ࢆྲྀࡾ
ࡾୖࡆࡿ㸬኱ᚄ
ᚄ㒊እ࿘㠃ࢆ
ࢆᣊ᮰ࡋ࡚ᑠ
ᑠᚄ㒊ඛ➃࡟
࡟ࢺࣝ
ࢡࢆ୚
୚࠼ࡓ࡜ࡁࡢ
ࡢᑠᚄ㒊ࡢᅇ
ᅇ㌿ゅࡢศᕸ
ᕸࢆ㸪ᅗ 5.10
0 ࡢ 3D FEM
M ࣔࢹࣝ࡜ᅗ
ᅗ 5.11 ࡢ 1D
D FEM
ࣔࢹࣝ
ࣝ࡟ࡘ࠸࡚ồ
ồࡵ࡚ẚ㍑ࢆ
ࢆ⾜ࡗࡓ㸬ᅗ
ᅗ 5.11 ࡢ 1D
D FEM ࣔࢹࣝ
࡛ࣝࡣᕥ➃࠿
࠿ࡽ 5 ␒┠ࡢ
ࡢせ⣲
㛗ࢆᘧ
ᘧ㸦5.10㸧࡛
࡛୚࠼ࡽࢀࡿ
ࡿ㛗ࡉ࡜ࡋ࡚
࡚࠸ࡿ㸬
Fixed
T
ᅗ 5.10 3D FEM
F
model of
o shaft with sstepped crosss-section for torsional
t
defllection analyssis
Fixed
T
ᅗ 5.11 1D FEM
F
model of
o shaft with sstepped crosss-section for torsional
t
defllection analyssis
72
D FEM ࣔࢹࣝ
࡛ࣝࡣ㸪㍈┤
┤ᚄኚ໬㒊᩿
᩿㠃ࡢᙎᛶኚ
ኚᙧࢆ⪃៖ࡋ
ࡋ࡞࠸ 1D Moodel 1 ࡜ᥦ᱌
᱌᪉ἲ
1D
࡟ࡼࡗ
ࡗ࡚㍈┤ᚄኚ
ኚ໬㒊᩿㠃ࡢ
ࡢᙎᛶኚᙧࢆ
ࢆ⪃៖ࡋࡓ 1D
1 Model 3 ࡢ 2 ࡘࡢࣔࢹ
ࢹࣝ࡟ࡘ࠸࡚
࡚ᅇ㌿
ゅࢆồ
ồࡵࡓ㸬ࡇࡇ
ࡇ࡛㸪ᑠᚄ㒊
㒊ඛ➃࡟୚࠼
࠼ࡿࢺࣝࢡࡣ T=100 N‚m ࡜ࡋࡓ㸬ᶓᙎ
ᙎᛶಀᩘࡣ㸪5.5.2
⠇࡜ྠ
ྠࡌ್ࢆ⏝࠸
࠸ࡓ㸬
ᅗ 5.12 ࡣྛࣔ
ࣔࢹࣝࡢᑠᚄ
ᚄ㒊ࡢᅇ㌿ゅ ࡢศᕸࢆ♧ࡋࡓࡶࡢ࡛࠶
࠶ࡿ㸬ྠᅗ࡟
࡟ࡣඛ➃㒊ࡢ
ࡢᣑ኱
ຍ࠼࡚ඛ➃㒊
㒊ࡢᅇ㌿ゅࡢ
ࡢ್ࢆซ౛ࡢ
ࡢ㡰࡟♧ࡋࡓ
ࡓ㸬3D FEM ࣔࢹࣝ࡟ᑐ
ᑐࡍࡿ 1D Mo
odel 1
ᅗ࡟ຍ
࡜ 1D
D Model 3 ࡢඛ➃㒊ࡢᅇ
ࡢ
ᅇ㌿ゅࡢㄗᕪ
ᕪࡣࡑࢀࡒࢀ
ࢀ㸪-1.46%࡜-0.47%
࡛࠶ࡾ
ࡾ㸪ᥦ᱌᪉ἲ
ἲ࡟ࡼ
ࡗ࡚㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃ࡢᙎᛶ
ᛶኚᙧࢆ⪃៖
៖ࡍࡿࡇ࡜࡟
࡟ࡼࡗ࡚ඛ➃
➃㒊ࡢᅇ㌿ゅ
ゅࡢㄗᕪࢆ኱
኱ᖜ࡟
పῶ࡛
࡛ࡁࡿࡇ࡜ࡀ
ࡀ☜ㄆࡉࢀࡓ
ࡓ㸬
T=100 (Nm
m)
Maximum
m Deflection
0.7479 d
deg
0.7369 ddeg
0.7444 ddeg
Fig.. 5.12
Defflection of shaaft with stepp
ped cross-secttion due to toorque
ᙜࡢゎᯒࣔࢹ
ࢹࣝ࡟ࡼࡿࡡ
ࡡࡌࡾ࣮ࣔࢻࡢᅛ᭷᣺ືᩘ
ᩘࡢホ౯
ᐇᶵ┦ᙜ
ᮏ⠇
⠇࡛ࡣ㸪ᅗ 5.13 ࡟♧ࡍ
ࡍࢱ࣮ࣅࣥ㸫Ⓨ
Ⓨ㟁ᶵ࣮ࣟࢱࡢᐇᶵ┦ᙜ
ᙜࣔࢹࣝࢆ⏝
⏝࠸࡚㸪㍈┤
┤ᚄኚ
໬㒊᩿
᩿㠃ࡢᙎᛶኚ
ኚᙧࢆᥦ᱌᪉
᪉ἲ࡟ࡼࡗ࡚
࡚⪃៖ࡋࡓሙ
ሙྜ㸦ᥦ᱌ἲ
ἲ㸧࡜⪃៖ࡋ
ࡋ࡞࠸ሙྜ㸦ᚑ᮶
ἲ㸧࡟
࡟ࡘ࠸࡚㸪ࡡ
ࡡࡌࡾ࣮ࣔࢻ
ࢻࡢᅛ᭷್ゎ
ゎᯒ⤖ᯝࢆẚ
ẚ㍑ࡍࡿ㸬
73
Tuurbine
Generator
Coupling
Exciter
E
Fig. 5.13
Torssional vibratioon analysis model
m
of turbiine – generatoor rotor
ࢹࣝࡣ㸪ࢱ࣮ࣅ࣮ࣥࣟࢱ ࡜Ⓨ㟁ᶵ࣮ࣟ
࣮ࢱࡀ࢝ࢵࣉ
ࣉࣜࣥࢢ࡛๛
๛⤖ྜࡉࢀࡓ
ࡓ1㍈
ᅗ 5.13 ࡢࣔࢹ
ࢱࡀ⠇Ⅼ␒ྕ
ྕ 6㸪45㸪67
7㸪99㸪107 ࡢ఩⨨ࡢ㍈ཷ
ཷ࡛ᨭᣢࡉࢀ
ࢀࡓᵓ㐀ࢆᣢ
ᣢࡗ࡚࠸ࡿ㸬ࢱ࣮
࣮ࣟࢱ
ࣅࣥ㸫Ⓨ㟁ᶵ࣮ࣟ
࣮ࢱࡣ㸪඲㛗
㛗 29,614mm
m㸪⠇Ⅼᩘ 10
08㸪㍈ࡢせ⣲
⣲ᩘ 107 ࡢᅇ
ᅇ㌿㍈࡜㸪ࣈ
ࣈ࣮ࣞ
ࡢࢹ࢕ࢫࢡせ
せ⣲࡛ᵓᡂࡉ
ࡉࢀ࡚࠸ࡿ㸬ỈⰍ࡛♧ࡋ
ࡋࡓ㸪せ⣲␒ྕ 4㸪7㸪9㸪21㸪28㸪34㸪39㸪
ࢻ➼ࡢ
41㸪443㸪50㸪53㸪59㸪62㸪68
6 㸪74㸪91㸪 97㸪102㸪105 ࡢせ⣲ࡣ
ࡣ㸪㍈┤ᚄኚ
ኚ໬㒊᩿㠃ࡢ
ࡢᙎᛶ
ኚᙧ࡟
࡟ࡼࡿࡡࡌࡾ๛ᛶపୗࢆ
ࢆ⪃៖ࡍ࡭ࡁ
ࡁせ⣲࡛࠶ࡿ
ࡿࡓࡵ㸪せ⣲
⣲㛗ࢆᘧ㸦5.110㸧࡛୚࠼ࡽ
ࡽࢀࡿ
㛗ࡉ࡜ࡋ࡚࠸ࡿ㸬௜㘓 A-5 ࡟ࢱ࣮ࣅࣥ 㸫Ⓨ㟁ᶵ࣮ࣟ
࣮ࢱࡢࡡࡌࡾ᣺ືゎᯒࣔ
ࣔࢹࣝࡢㅖඖ
ඖࢆ♧
ࡍ㸬
⾲ 5.3 ࡣ㸪ỈⰍ
Ⰽ࡛♧ࡋࡓせ
せ⣲ࡢࡡࡌࡾ ๛ᛶࢆ G Ipeeq㸦ᘧ㸦5.14㸧ཧ↷㸧࡜ࡍ
ࡍࡿᥦ᱌᪉ἲ
ἲ࡜㸪
ᚄኚ໬㒊ࡢࡡ
ࡡࡌࡾ๛ᛶప
పୗࢆ⪃៖ࡋ
ࡋ࡞࠸㏻ᖖࡢ
ࡢ᪉ἲ࡟ࡘ࠸
࠸࡚㸪ࡡࡌࡾ 1 ḟ㹼3 ḟࡢ
ࡢᅛ᭷
㍈┤ᚄ
᣺ືᩘ
ᩘࡢィ⟬⤖ᯝ
ᯝࢆ♧ࡋࡓࡶ
ࡶࡢ࡛࠶ࡿ㸬 ྠ⾲࡟ࡣ㸪ᥦ᱌᪉ἲ࡜
࡜㏻ᖖࡢ᪉ἲ
ἲࡢᕪ␗࡟ࡘ
ࡘ࠸࡚
ࡶ♧ࡋࡓ㸬ᥦ᱌ἲ
ἲ࡟ࡼࡿࡡࡌ
ࡌࡾ 1 ḟ࡜ 2 ḟࡢᅛ᭷᣺
᣺ືᩘࡣ㸪㏻
㏻ᖖࡢ᪉ἲࡼ ࡾ⣙ 2%ప࠸
࠸್࡛
㸬ཧ⪃࡟㸪ࡡ
ࡡࡌࡾ 1 ḟ㹼
㹼3 ḟࡢᅛ᭷
᭷࣮ࣔࢻࢆᅗ
ᅗ 5.14 ࡟♧ࡍ
ࡍ㸬
࠶ࡿ㸬
Table 5.3
Naturaal frequenciess of torsional modes
for turbbine – generattor rotor
Un
nit: Hz
Modee Proposed method
Ordinary metthod
1
16.8873
17.253
(2.25%)
2
132..38
134.94
(1.93%)
3
166..78
167.14
(0.21%)
74
1
0.8
1st
2nd
3rd
0.6
Deflection
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
Fig. 5.14
5
10
15
20
25
30
Position of nodes (m)
Eigenmodes of torsional vibration for turbine – generator rotor
ᅗ 5.13 ࡢゎᯒࣔࢹࣝ࡟㛵ࡍࡿ㸪ᥦ᱌᪉ἲ࡜㏻ᖖࡢ᪉ἲ࡟ࡼࡿ 1 ḟ࡜ 2 ḟࡢࡡࡌࡾ࣮ࣔࢻ
ࡢᅛ᭷್ゎᯒ⤖ᯝࡢᕪ␗ ⣙ 2%ࡀ㸪࡝ࡢ⛬ᗘࡢ኱ࡁࡉ࡛࠶ࡿ࠿ࢆホ౯ࡍࡿࡓࡵ࡟㸪ࡇࡢᕪ
␗ࢆつ᱁࡛♧ࡉࢀ࡚࠸ࡿ್࡜ẚ㍑ࡋࡓ㸬
ᅗ 5.15 ࡣ㸪50 MW ௨ୖࡢ㝣⏝ࢱ࣮ࣅࣥⓎ㟁ᶵ࣮ࣟࢱ࡟㛵ࡍࡿࡡࡌࡾ᣺ື ࡟㛵ࡍࡿタィ
ᣦ㔪࡟㛵ࡍࡿつ᱁ ISO 22266-1: 2009㸦International Organization for Standardization, 2009㸧࡛♧
ࡉࢀ࡚࠸ࡿ㸪ಖࡘ࡭ࡁ㸪ྠᮇ࿘Ἴᩘ࡜ࡡࡌࡾᅛ᭷᣺ືᩘ࡜ࡢᕪ␗㸪ࡍ࡞ࢃࡕ㞳ㄪࡢᗘྜ࠸
ࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬ྠつ᱁࡛ࡣ㸪A1㸪A2㸪B㸪C㸪D㸪E ࡢ್ࢆࡑࢀࡒࢀ㸪2.5%㸪2.5%㸪
1%㸪2.5%㸪1%㸪2%࡜ࡍࡿࡇ࡜ࡀ୍⯡ᣦ㔪࡜ࡋ࡚♧ࡉࢀ࡚࠸ࡿ㸬ࡉࡽ࡟㸪ࡡࡌࡾ᣺ືࡢゎ
ᯒࡲࡓࡣィ ࡀᖖ ࡛ᐇ᪋ࡉࢀࡿሙྜࡣ㸪1%ࡢవ⿱ࢆ㏣ຍ࡛ぢ㎸ࡴᚲせࡀ࠶ࡿࡇ࡜ࡀ♧ࡉ
ࢀ࡚࠸ࡿ㸬
75
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RUDVLPLODUVK
KDIWV\VWHP
)7HPSHUDWX
XUHHIIHFWV
7KLVFRPS
SHQVDWHVIRUWK
KHFKDQJHLQVVKDIWVWLIIQHVVLQWKRVHFDVHV
VZKHUHFDOFXXODWLRQVRU
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RSHUDWLQJWHP SHUDWXUH )LV]HURLIWHPSHUDWXUH HIIHFWVKDYH EHHQWDNHQLQWRDFFRXQW㻌
Fig. 5.15
Definition of torsionall frequency exclusion
e
zone (ISO 222666-1: 2009)
ᅗ 5.13 ࡢ࣮ࣟࢱࡢゎᯒࣔࢹ
ࢹࣝ࡟࠾࠸࡚
࡚⏕ࡌࡓᥦ᱌
᱌ἲ࡜㏻ᖖࡢ
ࡢ᪉ἲ࡜ࡢࡡ
ࡡࡌࡾ࣮ࣔࢻ
ࢻࡢᅛ
ゎᯒ⤖ᯝࡢᕪ
ᕪ␗ ⣙ 2%ࡣ㸪
ࡣ ISO 222666-1: 2009 ୍࡛⯡ᣦ㔪࡜
࡛
࡜ࡋ࡚♧ࡉࢀ
ࢀ࡚࠸ࡿ㸪⪃
⪃៖ࡍ
᭷್ゎ
࡭ࡁゎ
ゎᯒㄗᕪࡢ኱
኱ࡁࡉ 2.5%
%࡜ྠ⛬ᗘ࡛
࡛࠶ࡿ࡜ゝ࠼ࡿ㸬ࡇࡇ࡛ྲྀ
ྲྀࡾୖࡆࡓ㍈
㍈┤ᚄኚ໬㒊
㒊᩿㠃
ࡢᙎᛶ
ᛶኚᙧ࡟ࡼࡿ
ࡿࡡࡌࡾ๛ᛶ
ᛶపୗ࡟ຍ࠼
࠼࡚㸪↓ど࡛
࡛ࡁ࡞࠸኱ࡁ
ࡁࡉࡢࡡࡌࡾ ᣺ືゎᯒࣔࢹࣝ
ࡢࣔࢹ
ࢹࣝ໬ㄗᕪࡀ
ࡀᏑᅾࡍࡿሙ
ሙྜࡣ㸪ゎᯒ
ᯒㄗᕪࡀつ᱁
᱁୍࡛⯡ᣦ㔪
㔪࡜ࡋ࡚♧ࡉ ࢀࡓ್ࢆୖ
ୖᅇࡿ
ᜍࢀࡀ
ࡀ࠶ࡿ㸬
76
⤖ゝ
1 ḟඖᱱࡢ᭷㝈せ⣲ࣔࢹࣝ࡬ࡢ㐺⏝ࡀᐜ᫆࡞㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅ
ࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗࢆ⪃៖ࡋࡓࣔࢹࣝ໬᪉ἲ࡟ࡘ࠸᳨࡚ウࡋ㸪௨ୗࡢ⤖ㄽࢆᚓࡓ㸬
(1) ኱ᚄ㒊ࡀཌᯈ࡜ࡳ࡞ࡏࡿ㸪ᙎᛶኚᙧࢆ⪃៖ࡍ࡭ࡁ h/d0 ࡢ⠊ᅖ㸪ࡍ࡞ࢃࡕᥦ᱌᪉ἲࡢ㐺
⏝⠊ᅖࡣ㸪㍈┤ᚄẚ ȕ ࡟ࡼࡽࡎ࠾ࡼࡑ h/d0 • 0.2 ࡛࠶ࡿ㸬ࡇࢀࡣ᭤ࡆࡢሙྜ࡜ྠᵝ࡛࠶
ࡿ㸬
(2) ᑠᚄ㒊࡟㛗ࡉ 3ʌd/{64(1+0.3382ȕ3+0.0815ȕ5+0.1144ȕ6+0.0125ȕ7)}ࡢ㍈せ⣲ࢆタࡅ㸪ࡑࡢ㒊
ศࡢࡡࡌࡾ๛ᛶࢆ᭤ࡆࡢሙྜ࡜ྠᵝࡢᘧ㸦5.19㸧࡛୚࠼ࡽࢀࡿ➼౯࡞ࡡࡌࡾ๛ᛶ Ipeq ࡜
ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗ
ࢆ⾲⌧ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
2
4
2
2
3
I p1 I p 2 I p1 I p1 7 I p1 I p 2 6 I p1 I p 2 2 I p 2
I peq
2 I p1 I p 2
4
㸦5.19㸧
Ip1㸸኱ᚄ㍈ࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ (m4)
Ip2㸸ᑠᚄ㍈ࡢ᩿㠃 2 ḟᴟ࣮࣓ࣔࣥࢺ(m4)
(3) ኱ᚄ㒊ࡀཌᯈ࡜ࡳ࡞ࡏࡿ㸪ᙎᛶኚᙧࢆ⪃៖ࡍ࡭ࡁ⠊ᅖ࡛ࡣ㸪኱ᚄ㍈┤ᚄ d0 ࡀᑠᚄ㍈┤
ᚄ d ࡟ᑐࡋ࡚࠸ࡃࡽ኱ࡁࡃ࡞ࡗ࡚ࡶ㸪➼౯࡞ࡡࡌࡾ๛ᛶ GIpeq ࡣ᭤ࡆࡢሙྜ࡜ྠᵝ࡟ᑠ
ᚄ㍈ࡢࡡࡌࡾ๛ᛶ GIp2 ࡢ 0.4979 ಸࢆୗᅇࡽ࡞࠸㸬
(4) ᥦ᱌᪉ἲࢆ㐺⏝ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪ືⓗ⌧㇟࡛࠶ࡿᅛ᭷᣺ື≉ᛶࡔࡅ࡛࡞ࡃ㸪㟼ⓗ࡞
ࡡࡌࡾኚᙧ࡟ࡘ࠸࡚ࡶ㧗⢭ᗘ࡞ゎᯒ⤖ᯝࢆᚓࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
(5) ᐇᶵ┦ᙜࡢࢱ࣮ࣅࣥ㸫Ⓨ㟁ᶵ࣮ࣟࢱࡢࣔࢹࣝ࡟࠾ࡅࡿ㸪పḟ࣮ࣔࢻࡢᅛ᭷᣺ືᩘ࡟㛵
ࡍࡿ㏻ᖖࡢ᪉ἲ࡜ᥦ᱌᪉ἲࡢᕪ␗ࡣ㸪᭱኱࡛⣙ 2%࡛࠶ࡗࡓ㸬ࡇࡢ್ࡣ㸪ISO 22266-1:
2009 ୍࡛⯡ᣦ㔪࡜ࡋ࡚♧ࡉࢀ࡚࠸ࡿ㸪⪃៖ࡍ࡭ࡁゎᯒㄗᕪࡢ኱ࡁࡉ 2.5%࡜ྠ⛬ᗘ࡛
࠶ࡿ㸬
ཧ⪃ᩥ⊩
୰ཎ ୍㑻, ᛂ⏝ᙎᛶᏛ, ᐇᩍฟ∧ (1977), p. 91.
୰ἑ ୍, 㛗ᒇ ஧㑻, ຍ⸨ ༤, ᭱᪂ᶵᲔᕤᏛㅮᗙ ᮦᩱຊᏛ, ⏘ᴗᅗ᭩ (1973), p. 46.
Sneddon, I. N., The Reissner-Sagoci problem, Proceedings of The Glasgow Mathematical
Association , Vol. 7, No. 3 (1966)
ISO 22266-1: 2009, Mechanical vibration - torsional vibration of rotating machinery - part 1:
Land-based steam and gas turbine generator sets in excess of 50 MW, International
Organization for Standardization (2009).
77
➨ ❶ ᣺ືヨ㦂⤖ᯝ࡟ᇶ࡙ࡃ㍈᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬
⥴ゝ
࣮ࣟࢱࡢືⓗᣲືࡢண ⢭ᗘࡣ㸪࣮ࣟࢱࡢ᣺ືゎᯒࣔࢹࣝ㸪㍈ཷࡸࢩ࣮ࣝ㒊࡞࡝ࡢᨭᣢ
㒊ࡢ๛ᛶ࡜ῶ⾶≉ᛶ㸪࠾ࡼࡧእຊࡢ⢭ᗘ࡟౫Ꮡࡍࡿ㸬≉࡟㸪㧗㏿ᅇ㌿ࡢ࣮ࣟࢱ࡛ࡣᅇ㌿ᩘ
౫Ꮡ㡯ࢆྵࡵࡓ᣺ືゎᯒࣔࢹࣝࡢ⢭ᗘࡀ㔜せ࡜࡞ࡿ㸬ࡑࡢࡓࡵ㸪㛤Ⓨẁ㝵ࡢ࣮ࣟࢱ࡛ࡣࡋ
ࡤࡋࡤ㸪ᐇᶵ࣮ࣟࢱࡢ᣺ືヨ㦂࡛ᚓࡽࢀࡿ㸪࿘Ἴᩘᛂ⟅㛵ᩘ㸪ᅛ᭷᣺ື㸪࠾ࡼࡧᅛ᭷࣮ࣔ
ࢻ࡞࡝ࡢᙎᛶ≉ᛶࡢ࠺ࡕ㸪࿘Ἴᩘᛂ⟅㛵ᩘ࡟ᇶ࡙࠸࡚᣺ືゎᯒࣔࢹࣝࢆ㧗⢭ᗘ໬ࡍࡿ᪉ἲ
㸦Okuma and Heylen, 2000㸧ࡸ㸪ᅛ᭷᣺ືᩘ࡜ᅛ᭷࣮ࣔࢻ࡟ᇶ࡙࠸࡚ゎᯒ⪅ࡀ㑅ᐃࡋࡓᮦᩱ
≉ᛶ್ࢆ᭱㐺໬ࡍࡿ᪉ἲ࡞࡝࡟ࡼࡗ࡚㸪᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬ࡀ࡞ࡉࢀࡿ㸬ࡋ࠿ࡋ㸪
ᅛ᭷᣺ືᩘ࡜ᅛ᭷࣮ࣔࢻࡢࡳ࡟╔┠ࡋ࡚㧗⢭ᗘ໬ࡉࢀࡓ᣺ືゎᯒࣔࢹ࡛ࣝࡣ㸪Ẽయ㍈ཷࡸ
☢Ẽ㍈ཷ࡛ᨭᣢࡉࢀࡓ࣮ࣟࢱ࡞࡝࡛㔜せ࡜࡞ࡿ๛య࣮ࣟࢱ㸦㍈ࡢᙎᛶኚᙧࡀ↓ど࡛ࡁࡿࣟ
࣮ࢱ㸧࡜ࡋ࡚ᢅ࠺ࡇ࡜ࡀ࡛ࡁࡿపᅇ㌿ᩘᇦࡢᅛ᭷᣺ືᩘ࡜ᅛ᭷࣮ࣔࢻࡢ⢭ᗘࡣಖドࡉࢀ࡞
࠸㸬ࡉࡽ࡟㸪ᚋ⪅ࡢ᪉ἲ࡛ࡣࢺࣛ࢖࢔ࣥࢻ࢚࣮ࣛࢆせࡍࡿࡓࡵ㸪㧗⢭ᗘ໬ࡢసᴗຠ⋡ࡣᚲ
ࡎࡋࡶⰋ࠸࡜ࡣ࠸࠼࡞࠸㸬
ࡑࡇ࡛㸪ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ࡛ࣔࢹࣝ໬ࡉࢀࡿ㍈࡜๛యࢹ࢕ࢫࢡ࡛ᵓᡂࡉࢀࡿ࣮ࣟࢱࡢ᣺
ືゎᯒࣔࢹࣝ࡟ࡘ࠸࡚㸪㠀ᅇ㌿᫬ࡢ᣺ືヨ㦂࡛ᚓࡽࢀࡿ㉁㔞㸪㔜ᚰ఩⨨㸪័ᛶ࣮࣓ࣔࣥࢺ
㸦ࡇࢀࡽࢆ๛య≉ᛶ࡜࿧ࡪ㸧࡜㸪ᅛ᭷᣺ືᩘ㸪ᅛ᭷࣮ࣔࢻ㸦ࡇࢀࡽࢆᙎᛶ≉ᛶ࡜࿧ࡪ㸧࡟
㛵ࡍࡿ᝟ሗ࠿ࡽ㸪ㄗᕪࢆྵࢇࡔ඲࡚ࡢᮦᩱ≉ᛶ್ࢆྠ᫬࡟㧗⢭ᗘ໬ࡋ࡚㸪᣺ື᪉⛬ᘧ࡟࠾
ࡅࡿ඲࡚ࡢ≉ᛶ⾜ิ㸦㉁㔞⾜ิ㸪ῶ⾶⾜ิ㸪๛ᛶ⾜ิ㸪࠾ࡼࡧࢪࣕ࢖ࣟ⾜ิ㸧ࢆྠᐃࡍࡿ
᪉ἲࢆ♧ࡍ㸦㛗Ụ, ᚋ⸨, 2013㸧㸬
6.2 ⠇࡛ࡣ≉ᛶ⾜ิࡢྠᐃ࡟ࡘ࠸࡚ᴫㄝࡍࡿ㸬
6.3 ⠇࡛ࡣ᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ≉ᛶ⾜ิࡢྠᐃ᪉ἲࢆᴫㄝࡍࡿ㸬
6.4 ⠇࡛ࡣᮍ▱ࣃ࣓࣮ࣛࢱࡢලయⓗ࡞ྠᐃᡭ㡰ࢆ♧ࡍ㸬
6.5 ⠇࡛ࡣ㸪๛య≉ᛶ࡜ᙎᛶ≉ᛶ࡟㛵ࡍࡿᶍᨃࡢᐇ ್࡟ࡼࡿㄗᕪࢆᣢࡘゎᯒࣔࢹࣝࡢ㧗
⢭ᗘ໬ࡢලయ౛࡜ࡋ࡚㸪4 ẁࡢ㐲ᚰᅽ⦰ᶵ࣮ࣟࢱࡢࣔࢹࣝࡢ≉ᛶ⾜ิࢆྠᐃࡋࡓ⤖ᯝࢆ♧ࡍ㸬
≉ᛶ⾜ิࡢྠᐃࡢᴫせ
➨ 2 ❶࡛㏙࡭ࡓᵝ࡟㸪㍈せ⣲ࡢ㉁㔞ಀᩘ Įm㸪័ᛶಀᩘ Įi㸪ࡏࢇ᩿ಀᩘ IJ㸪᭤ࡆ๛ᛶ EI㸪
㛗ࡉ le㸪࠾ࡼࡧẚ౛ῶ⾶ᐃᩘ Ȗ㸦ࡇࢀࡽࢆ⥲⛠ࡋ࡚㍈ࡢᵓᡂࣃ࣓࣮ࣛࢱ࡜࿧ࡪࡇ࡜࡟ࡍࡿ㸧
࡜㸪ࢹ࢕ࢫࢡࡢ㉁㔞 mdisc㸪┤ᚄ㍈࿘ࡾࡢ័ᛶ࣮࣓ࣔࣥࢺ Jdisc㸪࠾ࡼࡧཌࡉ tdisc㸦ࡇࢀࡽࢆ⥲
⛠ࡋ࡚ࢹ࢕ࢫࢡࡢᵓᡂࣃ࣓࣮ࣛࢱ࡜࿧ࡪࡇ࡜࡟ࡍࡿ㸧ࡀᐃࡲࢀࡤ㸪࣮ࣟࢱ඲యࡢ᣺ື᪉⛬
ᘧ࡟࠾ࡅࡿ඲࡚ࡢ≉ᛶ⾜ิ㸪ࡍ࡞ࢃࡕ㉁㔞⾜ิ[Mr]㸪๛ᛶ⾜ิ[Kr]㸪ῶ⾶⾜ิ[Cr]㸪࠾ࡼࡧࢪ
ࣕ࢖ࣟ⾜ิ[Gr]ࢆᐃࡵࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
⮬⏤ᨭᣢࡉࢀࡓ㠀ᅇ㌿᫬ࡢ࣮ࣟࢱ࡟࠾࠸࡚㸪㉁㔞㸪๛ᛶ㸪࠾ࡼࡧῶ⾶ࡢྛ⾜ิࡀᐃࡲࢀ
ࡤ㸪࣮ࣟࢱࡢ㉁㔞㸪㔜ᚰ఩⨨㸪࣮ࣟࢱࡢ㔜ᚰࢆ㏻ࡾᅇ㌿㍈࡟┤஺ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ࣮ࣔ
࣓ࣥࢺ㸪࠾ࡼࡧ࣮ࣟࢱࡢᅇ㌿㍈࿘ࡾࡢᴟ័ᛶ࣮࣓ࣔࣥࢺ㸦ࡇࢀࡽࢆ⥲⛠ࡋ࡚๛య≉ᛶ࡜࿧
78
ࡪࡇ࡜࡟ࡍࡿ㸧࡜ᅛ᭷᣺ືᩘ࠾ࡼࡧᅛ᭷࣮ࣔࢻ㸦ࡇࢀࡽࢆ⥲⛠ࡋ࡚ᙎᛶ≉ᛶ࡜࿧ࡪࡇ࡜࡟
ࡍࡿ㸧ࡀ୍ព࡟ᐃࡲࡿ㸬㏫࡟㸪࣮ࣟࢱࡢᑍἲ࡜ᙧ≧ࡀỴࡲࡗ࡚࠸ࢀࡤ㸪๛య≉ᛶ࡜ᙎᛶ≉
ᛶ࠿ࡽ㸪㉁㔞㸪๛ᛶ㸪࠾ࡼࡧῶ⾶ࡢྛ⾜ิ࡟ຍ࠼࡚࣮ࣟࢱࡢࢪࣕ࢖ࣟ⾜ิ࡜๛ᛶ⾜ิࡢᅇ
㌿㏿ᗘ౫Ꮡ㡯ࢆ୍ព࡟ᐃࡵࡿࡇ࡜ࡀ࡛ࡁࡿ㸬ࡇࢀࡀ⌮ㄽୖࡢ୺ᖿࢆ࡞ࡍ㒊ศ࡛࠶ࡾ㸪࣮ࣟ
ࢱࡢ๛య≉ᛶ࡜ᙎᛶ≉ᛶࡀᐇ ྍ⬟࡛࠶ࡿሙྜࡣᐇ㦂࡟ᇶ࡙࠸ࡓ࣮ࣟࢱࡢ≉ᛶ⾜ิࡢྠᐃ
ࢆᐇ⌧ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
௨ୗ࡟㸪᣺ືゎᯒࣔࢹࣝࡢ⠇Ⅼ࡜ィ Ⅼࡀྠ୍ࡢሙྜ࡟ࡘ࠸࡚㸪≉ᛶ⾜ิࡢྠᐃ⌮ㄽࡢ
ヲ⣽ࢆ♧ࡍ㸬
≉ᛶ⾜ิࡢྠᐃ᪉ἲ
᣺ືゎᯒࣔࢹࣝࡢ඲࡚ࡢ⠇Ⅼࢆィ Ⅼ࡜ࡍࡿࡇ࡜ࡀ࡛ࡁࡿሙྜࡣ㸪㍈せ⣲ࡢ㛗ࡉ le㸪እᚄ
de㸪࠾ࡼࡧẚ౛ῶ⾶ᐃᩘ Ȗ ࡜ࢹ࢕ࢫࢡࡢ㉁㔞 mdisc ࡜ཌࡉ tdisc ࡀ᪤▱࡛࠶ࢀࡤ㸪ᐇ ࡉࢀࡓࣟ
࣮ࢱࡢ๛య≉ᛶ࡜ᙎᛶ≉ᛶ࠿ࡽ㸪඲࡚ࡢ≉ᛶ⾜ิࢆྠᐃࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬ᮍ▱ࡢ㍈ࡢᵓ
ᡂࣃ࣓࣮ࣛࢱࡣ㸪ྛ㍈せ⣲ࡢ㉁㔞ಀᩘ Įm ࡜័ᛶಀᩘ Įi ࡢ࡝ࡕࡽ࠿୍᪉㸪ࡏࢇ᩿ಀᩘ IJ㸪࠾
ࡼࡧ᭤ࡆ๛ᛶ EI ࡢ 3 ࡘ࡛࠶ࡾ㸪ྠ୍ᮦᩱࡢ㍈࡛ࡣ㸪㍈඲య࡟ࡘ࠸࡚ࡇࢀࡽ 3 ࡘࡢࡳࢆồࡵ
ࢀࡤࡼ࠸㸬ᮍ▱ࡢ㍈ࡢᵓᡂࣃ࣓࣮ࣛࢱࢆồࡵࡿၥ㢟ࡣ㍈せ⣲ࡢᐦᗘ ȡ㸪࣏࢔ࢯࣥẚ Ȟ㸪࠾ࡼ
ࡧࣖࣥࢢ⋡ E ࢆồࡵࡿၥ㢟࡜➼౯࡛࠶ࡿ㸬ᮍ▱ࡢࢹ࢕ࢫࢡࡢᵓᡂࣃ࣓࣮ࣛࢱࡣྛࢹ࢕ࢫࢡ
ࡢ┤ᚄ㍈࿘ࡾࡢ័ᛶ࣮࣓ࣔࣥࢺ Jdisc ࡛࠶ࡿ㸬
ᐇ㝿ࡢࡶࡢ࡜୍⮴ࡍ࡭ࡁ࣮ࣟࢱࡢ๛య≉ᛶࡣ㸪඲㉁㔞 m㸪㔜ᚰ఩⨨ zCG㸪㔜ᚰࢆ㏻ࡾᅇ㌿
㍈࡟┤஺ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ࣮࣓ࣔࣥࢺ JCG ࡛࠶ࡾ㸪ᴟ័ᛶ࣮࣓ࣔࣥࢺ㸦ᅇ㌿㍈࿘ࡾࡢ୺័
ᛶ࣮࣓ࣔࣥࢺ㸧JpCG ࡣ㸪㍈࡜ࢹ࢕ࢫࢡࡢᵓᡂࣃ࣓࣮ࣛࢱ࡟ࡼࡗ࡚ᐃࡲࡿࡓࡵ⪃៖ࡍࡿᚲせࡣ
࡞࠸㸬
࣮ࣟࢱࡢ㉁㔞࡜ࢹ࢕ࢫࢡࡢ㉁㔞ࡣ⢭ᗘⰋࡃ ᐃྍ⬟࡛࠶ࡾ㸪㔜ᚰ఩⨨ zCG㸪㔜ᚰࢆ㏻ࡾᅇ
㌿㍈࡟┤஺ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ࣮࣓ࣔࣥࢺ JCG㸪ᅛ᭷᣺ືᩘ f㸪ᅛ᭷࣮ࣔࢻ ij㸪࠾ࡼࡧẚ౛ῶ
⾶ẚ Ȗ ࡣ㠀ᅇ㌿᫬ࡢ࣮ࣟࢱࡢ᣺ືヨ㦂࠿ࡽồࡵࡿࡇ࡜ࡀ࡛ࡁࡿ㸬࣮ࣟࢱࡢ๛య≉ᛶࡣ࿘Ἴᩘ
ᛂ⟅㛵ᩘ࠿ࡽồࡵࡿࡇ࡜ࡀ࡛ࡁ㸦Fareed, Wahl, 2001, ኱⇃, Heylen, 2000㸧㸪࣮ࣟࢱࡢᙎᛶ≉ᛶ
ࡶ࿘Ἴᩘᛂ⟅㛵ᩘࢆ⏝࠸ࡿ୍⯡ⓗ࡞ᐇ㦂࣮ࣔࢻゎᯒ࡟ࡼࡗ࡚ồࡵࡿࡇ࡜ࡀ࡛ࡁࡿ㸬࡞࠾㸪
ᅛ᭷࣮ࣔࢻ࡟ࡘ࠸࡚ࡣィ ࡢᐜ᫆ࡉ࡜ィ ⢭ᗘࢆ⪃៖ࡋ࡚୪㐍⮬⏤ᗘࡢࡳࢆ⪃࠼ࢀࡤⰋ࠸㸬
ᮍ▱ࣃ࣓࣮ࣛࢱࡢྠᐃᡭ㡰
᪤▱ࡢ㍈࡜ࢹ࢕ࢫࢡࡢᵓᡂࣃ࣓࣮ࣛࢱ࡜ᐇ ࡉࢀࡓ๛య≉ᛶ࡜ᙎᛶ≉ᛶ࠿ࡽ㸪ᮍ▱ࡢ㍈
࡜ࢹ࢕ࢫࢡࡢᵓᡂࣃ࣓࣮ࣛࢱࢆồࡵࡿᡭ㡰ࢆ௨ୗ࡟♧ࡍ㸬
᣺ືゎᯒࣔࢹࣝ࠿ࡽồࡵࡓ๛య≉ᛶ㸦㉁㔞㸪㔜ᚰ఩⨨㸪࠾ࡼࡧ㔜ᚰࢆ㏻ࡾᅇ㌿㍈࡟┤஺
ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ࣮࣓ࣔࣥࢺ㸧࡜ᙎᛶ≉ᛶ㸦ᅛ᭷᣺ືᩘ㸪ᅛ᭷࣮ࣔࢻ㸧ࡢ್࡜㸪㠀ᅇ㌿
᫬ࡢ࣮ࣟࢱࡢ᣺ືヨ㦂࠿ࡽồࡵࡓࡇࢀࡽࡢ್࡜ࡢㄗᕪࢆồࡵ㸪ㄗᕪࡢ஧஌࿴ࡀ᭱ᑠ࡜࡞ࡿ
ࡼ࠺࡞㍈ࡢ㉁㔞ಀᩘ Įm ࡲࡓࡣ័ᛶಀᩘ Įi㸪ࡏࢇ᩿ಀᩘ IJ㸪᭤ࡆ๛ᛶ EI㸦ᡈ࠸ࡣᐦᗘ ȡ㸪࣏࢔
79
ࢯࣥẚ Ȟ㸪ࣖࣥࢢ⋡ E㸧࡜㸪ࢹ࢕ࢫࢡࡢ┤ᚄ㍈࿘ࡾࡢ័ᛶ࣮࣓ࣔࣥࢺ Jdisc ࢆồࡵ㸪≉ᛶ⾜ิ
ࢆ෌ᵓᡂࡍࡿࡇ࡜࡟ࡼࡗ࡚ྠᐃࡀ᏶஢ࡍࡿ㸬ྠ୍ᮦᩱࡢ㍈࡛ࡣ㸪ᮍ▱ኚᩘࡣ 3+ ndisc ಶ㸦ndisc
ࡣࢹ࢕ࢫࢡࡢᩘ㸧࡛࠶ࡾ㸪ㄗᕪ㛵ᩘࡣ 3+ nf1+ nf2 ×n ಶ㸦nf1 ࡣᅛ᭷᣺ືᩘࡢᩘ㸪nf2 ࡣᅛ᭷
࣮ࣔࢻࡢᩘ㸪n ࡣ⠇Ⅼᩘ㸧タᐃࡍࡿࡇ࡜ࡀ࡛ࡁ㸪ndisc ࡣ n ௨ୗ࡛࠶ࡿࡓࡵ 3+ ndisc<3+ nf1+ nf2 ×n
࡜࡞ࡾ㸪᭱ᑠ஧஌ゎࡀᐃࡲࡿ㸬࡞࠾㸪ᐇ㦂࡛ࡣ㸪ᅛ᭷᣺ືᩘࡣ⢭ᗘⰋࡃồࡵࡿࡇ࡜ࡀ࡛ࡁ
ࡿࡀᅛ᭷࣮ࣔࢻࡣᅛ᭷᣺ືᩘ࡟ẚ࡭ࡿ࡜⢭ᗘⰋࡃồࡵࡿࡇ࡜ࡀ࡛ࡁ࡞࠸ሙྜࡀ࠶ࡿ㸬ࡑࡇ
࡛㸪ㄗᕪ㛵ᩘࡢཧ↷್࡜ࡋ࡚᥇⏝ࡍࡿᅛ᭷࣮ࣔࢻ࡜ࡋ࡚⢭ᗘࡢⰋ࠸ࡶࡢࡢࡳࢆ㑅ᢥࡍࡿࡇ
࡜ࢆ⪃៖ࡋ࡚㸪ᅛ᭷᣺ືᩘ࡟㛵ࡍࡿㄗᕪ㛵ᩘࡢᩘ nf1 ࡜ᅛ᭷࣮ࣔࢻ࡟㛵ࡍࡿㄗᕪ㛵ᩘࡢᩘ
nf2 ࢆศࡅ࡚࠸ࡿ㸬
ᐇ ್࡟ᑐࡍࡿ᣺ືゎᯒࣔࢹࣝ࡟ࡼࡿィ⟬್ࡢㄗᕪ㛵ᩘࢆ㸪௨ୗ࡟♧ࡍࡼ࠺࡟ᐇ ್࡟
ᑐࡍࡿẚ࡜ࡋ࡚ᐃ⩏ࡍࡿ㸬ῧ࠼Ꮠࡢ c ࡜ m ࡣࡑࢀࡒࢀィ⟬್㸦calculated value㸧࡜ᐇ ್
㸦measured value㸧ࢆ♧ࡍࡶࡢ࡜ࡍࡿ㸬࡞࠾㸪ᮏㄽᩥ࡛ࡣ⌮ㄽⓗጇᙜᛶࡢࡳ࡟╔┠ࡋ㸪๛య
≉ᛶ࡜ᙎᛶ≉ᛶࡢᐇ ࢹ࣮ࢱ࡟ྵࡲࢀࡿㄗᕪࢆ⪃៖ࡋ࡞࠸ࡶࡢ࡜ࡍࡿ㸬
㸦1㸧㉁㔞㸦࣮ࣟࢱࡢ඲㉁㔞㸧mR (kg)࡟㛵ࡍࡿㄗᕪ㛵ᩘ㸪1 ಶ
em
mRc mRm
(6.1)
mRm
㸦2㸧㔜ᚰ఩⨨㸦㍈᪉ྥࡢ㔜ᚰ఩⨨㸧zCG (m) ࡟㛵ࡍࡿㄗᕪ㛵ᩘ㸪1 ಶ
e z CG
z CG c z CG m
(6.2)
z CG m
㸦3㸧㔜ᚰࢆ㏻ࡾᅇ㌿㍈࡟┤஺ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ࣮࣓ࣔࣥࢺ JCG (kgm2) ࡟㛵ࡍࡿㄗᕪ㛵ᩘ㸪
1ಶ
eJ CG
J CGc J CGm
(6.3)
J CGm
㸦4㸧ᅛ᭷᣺ືᩘ f (Hz) ࡟㛵ࡍࡿㄗᕪ㛵ᩘ㸪 nf1 ಶ
80
^e ` ^e
f
f1
, e f 2 ,..., e fn f 1
T
­° f f
f cn f mn f 1 ½°
c1
m1 f c 2 f m 2
,
,..., f 1
®
¾ (6.4)
f m2
f mn f 1
°̄ f m1
°¿
`
T
㸦5㸧ᅛ᭷࣮ࣔࢻ ĭ ࡟㛵ࡍࡿㄗᕪ㛵ᩘ㸪 nf2×n ಶ
^e) ` ^^e)1`, ^e) 2 `,..., ^e)n
``
T
f2
^
`^
T
T
­
­° ) cn ) mn
°­ ^) c1` ^) m1`½ ­ ^) c 2 ` ^) m 2 `½
f2
f2
¾ ,®
¾ ,..., ®
®®
^
`
^
`
)
)
)
°̄
m2
m1
mn f 2
¿ ¯
¿
°¯
¯
^
`
`½° ½°
T
T
(6.5)
¾ ¾
°¿ °
¿
3+ ndisc ಶࡢᮍ▱ኚᩘࢆタィኚᩘ x ࡜ࡋ㸪ୖ㏙ࡢㄗᕪ㛵ᩘ࡟ᐇ ್ࡢಙ㢗ᛶࢆ⪃៖ࡋࡓ㔜
ࡳࢆ௜ࡅࡓࡶࡢࡢ஧஌࿴ࢆ┠ⓗ㛵ᩘ g(x)࡜ࡋ࡚㸪g(x)ࢆ᭱ᑠ໬ࡍࡿ㠀⥺ᙧ᭱ᑠ஧஌ၥ㢟࡬ࡢ
ᐃᘧ໬ࢆ⾜࠺㸬┠ⓗ㛵ᩘ g(x)ࢆḟᘧ࡛ᐃ⩏ࡍࡿ㸬
g x ne
¦ ^W e x `
2
i i
(6.6)
i 1
^ ` , ^e ` }
e i (x)㸸i ␒┠ࡢㄗᕪ㛵ᩘ㸪i=1, 2, …, ne㸪 ^ex ` {em , ezCG , eJCG , e f
Wi㸸i ␒┠ࡢ㔜ࡳಀᩘ㸪i=1, 2, …, ne㸪 ^W ` {W1 ,W2 ,...,Wne }T
ne㸸ㄗᕪ㛵ᩘࡢᩘ㸪ne=3+ nf1+ nf2 ×n
T
)
T T
ࡇࡇ࡛㸪㔜ࡳಀᩘ Wi ࡢ್ࡣ㸪ࡑࢀࡒࢀࡢㄗᕪ㛵ᩘ࡟ࡘ࠸࡚㸪ᐇ ್ࡢಙ㢗ᛶࡸ㏲ḟィ⟬ࡢ
཰᮰ᛶ࡞࡝ࢆ⪃៖ࡋ࡚Ỵᐃࡍࡿ㸬
௨ୖࡢ‽ഛࡢୗ㸪Trust Region Reflective Newton Method㸦Coleman and Li, 1996㸧࡞࡝ࡢไ
⣙᮲௳௜ࡁ᭱ᑠ஧஌ἲࡸ Simplex Search Method㸦Lagarias, et al., 1998㸧࡞࡝ࡢไ⣙࡞ࡋ᭱ᑠ
஧஌ἲࢆ㐺⏝ࡋ࡚㸪┠ⓗ㛵ᩘ g(x)ࢆ᭱ᑠ໬ࡍࡿᮍ▱ኚᩘࡢ᭱㐺್ࢆồࡵࡿࡇ࡜࡟ࡼࡗ࡚㸪᭱
㐺໬ࣃ࣓࣮ࣛࢱࡢ㑅ᐃ࡟ゎᯒ⪅ࡢពᛮࡀ௓ᅾࡋ࡞࠸᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬ࢆᐇ⌧ࡍࡿ
ࡇ࡜ࡀ࡛ࡁࡿ㸬࡞࠾㸪ไ⣙᮲௳ࡢ᭷↓࡟㛵ࢃࡽࡎ㸪᭱ᑠ஧஌ゎࢆᚓࡿࡓࡵ࡟㏲ḟィ⟬ᅇᩘ
ࡸチᐜㄗᕪ࡞࡝ࡢ᮲௳ࢆணࡵᣦᐃࡋ࡚࠾ࡃᚲせࡀ࠶ࡿ㸬ࡑࡢ௚࡟㸪ไ⣙᮲௳௜ࡁ᭱ᑠ஧஌
ἲ࡛ࡣ᥎ ࡉࢀࡿᮍ▱ኚᩘࡢㄗᕪ⠊ᅖࢆไ⣙᮲௳㸦ୖୗ㝈್㸧࡜ࡋ࡚஦๓࡟タᐃࡍࡿᚲせ
ࡀ࠶ࡾ㸪ไ⣙࡞ࡋ᭱ᑠ஧஌ἲ࡛ࡣᮍ▱ኚᩘࡢ᥎ᐃㄗᕪ࡜ẚ㍑ࡍࡿ࡞࡝ࡋ࡚ᚓࡽࢀࡓゎࡢጇ
ᙜᛶࢆ☜ㄆࡍࡿᚲせࡀ࠶ࡿ㸬
ᅗ 6.1 ࡣ㸪ไ⣙᮲௳௜ࡁ᭱ᑠ஧஌ἲ࡟ࡼࡿタィኚᩘࡢ᭱㐺໬ࡢࣇ࣮ࣟࢆ♧ࡋࡓࡶࡢ࡛࠶ࡿ㸬
81
(8)
(13)
(1)
(2)
(5)
((6)
(3)
(7)
(4)
(12)
(9)
(10)
(11)
*11 Design Variablees:
M
Mass coefficients Įm, Inertia coefficcients Įi, Shear cooefficients IJ, Bend
ding siffness EI, and
a Moments of iinertia
aabout diameter off discs Jdisc
orr Density of shaft ȡ, Poisson ratio Ȟ,
Ȟ Young's modullus E, and Momen
nts of inertia abou
ut diameter of disscs Jdisc
*22 Rigid & Elastic Properties:
Riigid properties: Total
T
mass of rotorr mR, Center of grravity zCG, Princip
pal moment of in
nertia JCG
Ellastic properties: Natural
N
frequencies f, Eigenmodess ĭ
*33 niter: number of iterations
Fig. 6.1
6
Flow chhart of optimiizing design valiables
v
by least squuares method with constraiint
ୗ࡟ᅗ 6.1 ࡢࣇ࣮ࣟࢆㄝ
ࡢ
ㄝ᫂ࡍࡿ㸬
௨ୗ
82
(1) ࣮ࣟࢱࡢ඲㉁㔞 mRm ࢆ ᐃࡍࡿ㸬
(2) ⮬⏤ᨭᣢࢆᶍᨃࡋࡓ≧ែࡢ࣮ࣟࢱࡢ᣺ືヨ㦂ࢆ⾜ࡗ࡚࿘Ἴᩘᛂ⟅㛵ᩘࢆィ ࡍࡿ㸬
(3) ࿘Ἴᩘᛂ⟅㛵ᩘ࠿ࡽ㸪㔜ᚰ఩⨨ zCGm ࡜㔜ᚰࢆ㏻ࡾᅇ㌿㍈࡟┤஺ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ
࣮࣓ࣔࣥࢺ JCGm ࢆồࡵࡿ㸬
(4) ࿘Ἴᩘᛂ⟅㛵ᩘ࠿ࡽ㸪ᅛ᭷᣺ືᩘ fm ࡜ᅛ᭷࣮ࣔࢻ ĭm ࢆồࡵࡿ㸦ᐇ㦂࣮ࣔࢻゎᯒ㸧㸬
(5) ࣮ࣟࢱࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝࢆసᡂࡋ㸪タィኚᩘ㸦ྛ㍈せ⣲ࡢ㉁㔞ಀᩘ Įm ࡲࡓࡣ័
ᛶಀᩘ Įi㸪ࡏࢇ᩿ಀᩘ IJ㸪᭤ࡆ๛ᛶ EI㸪ྛࢹ࢕ࢫࢡせ⣲ࡢ┤ᚄ㍈࿘ࡾࡢ័ᛶ࣮࣓ࣔࣥ
ࢺ Jdisc㸧ࢆồࡵࡿ㸬
(6) ࣮ࣟࢱࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝࡢ඲㉁㔞 mRc ࡜㔜ᚰ఩⨨ zCGc ࢆồࡵࡿ㸬
(7) ᅛ᭷್ゎᯒࢆ⾜ࡗ࡚ᅛ᭷᣺ືᩘ fc ࡜ᅛ᭷࣮ࣔࢻ ĭc ࢆồࡵࡿ㸬
(8) 㔜ࡳಀᩘ W㸪┠ⓗ㛵ᩘࡢチᐜ್㸪᭱኱㏲ḟィ⟬ᅇᩘ㸪タィኚᩘࡢୖୗ㝈್ࢆỴࡵࡿ㸬
(9) ㉁㔞 mR㸪㔜ᚰ఩⨨ zCG㸪
㔜ᚰࢆ㏻ࡾᅇ㌿㍈࡟┤஺ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ࣮࣓ࣔࣥࢺ JCG㸪
ᅛ᭷᣺ືᩘ f㸪ᅛ᭷࣮ࣔࢻ ĭ ࡟㛵ࡍࡿㄗᕪ㛵ᩘ࡜㔜ࡳಀᩘ W ࠿ࡽ㸪ᘧ㸦6.6㸧ࡢ┠ⓗ
㛵ᩘ g(x)ࢆタᐃࡍࡿ㸬
(10) ┠ⓗ㛵ᩘ g(x)ࡢ್ࡀチᐜ್ᮍ‶࡛࠶ࡿ࠿ࢆุᐃࡍࡿ㸬
(11) ┠ⓗ㛵ᩘ g(x)ࡢ್ࡀチᐜ್௨ୖ࡛࠶ࢀࡤ㸪㏲ḟィ⟬ᅇᩘࡀ᭱኱㏲ḟィ⟬ᅇᩘᮍ‶࡛࠶
ࡿ࠿ࢆุᐃࡍࡿ㸬
(12) ㏲ḟィ⟬ᅇᩘࡀ㏲ḟィ⟬ᅇᩘᮍ‶࡛࠶ࢀࡤ㸪ୖୗ㝈್ෆ࡛タィኚᩘࢆኚ໬ࡉࡏࡿ㸬
(13) ᪂ࡓ࡞タィኚᩘ࠿ࡽ㸪࣮ࣟࢱࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝࢆồࡵࡿ㸬
(14) ௨ୗ㸪(6)㸪(7)㸪(9)㸪(10)㸪(11)㸪(12)㸪(13)ࡢᡭ㡰ࢆ㸪┠ⓗ㛵ᩘࡀチᐜ್ᮍ‶࡜࡞ࡿ
ࡲ࡛㸪ᡈ࠸ࡣ㏲ḟィ⟬ᅇᩘࡀ᭱኱㏲ḟィ⟬ᅇᩘ௨ୖ࡜࡞ࡿࡲ࡛⧞ࡾ㏉ࡋ㸪タᐃࡋࡓ
᮲௳࡟࠾࠸࡚᭱㐺࡞࣮ࣟࢱࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝࢆồࡵࡿ㸬
๛య≉ᛶ࡜ᙎᛶ≉ᛶ࡟㛵ࡍࡿᶍᨃࡢᐇ ್࡟ࡼࡿㄗᕪࢆᣢࡘゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬
ࡇࡇ࡛ࡣ㸪㧗⢭ᗘ໬ࡢᑐ㇟࡜ࡋ࡚ 4 ẁࡢ㐲ᚰᅽ⦰ᶵ࣮ࣟࢱࡢࣔࢹࣝࢆᢅ࠸㸪᭤ࡆ᣺ືࡢ
ࡳ࡟╔┠ࡍࡿ㸬➨ 3 ❶࡛♧ࡋࡓ㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶ
పୗ࡟ࡘ࠸࡚ࡣ↓どࡍࡿ㸬
᣺ືゎᯒࣔࢹࣝ
㐲ᚰᅽ⦰ᶵ࣮ࣟࢱࡢ᣺ືゎᯒࣔࢹࣝࢆᅗ 6.2 ࡟♧ࡍ㸬࣮ࣟࢱࡣ 19 ࡢ⠇Ⅼࢆᣢࡘ㛗ࡉ 350mm
ࡢ㍈࡜⠇Ⅼ 7, 9, 11, 13 ࡟ྲྀࡾ௜ࡅࡽࢀࡓ 4 ᯛࡢࢹ࢕ࢫࢡ࡛ᵓᡂࡉࢀ࡚࠾ࡾ㸪㍈࡜ࢹ࢕ࢫࢡࡣ
ࡑࢀࡒࢀ⾲ 6.1 ࡜⾲ 6.2 ࡟♧ࡍᑍἲࢆᣢࡘ㸬㍈ࡢᮦᩱ≉ᛶ್ࡣᐦᗘ 7800kg/m3㸪ࣖࣥࢢ⋡
211GPa㸪࣏࢔ࢯࣥẚ 0.3㸦ᶓᙎᛶಀᩘ 81.154GPa㸧࡛࠶ࡿ㸬ࢹ࢕ࢫࢡࡢᐦᗘࡣࢹ࢕ࢫࢡ 1 ࠿
ࡽ㡰࡟ࡑࢀࡒࢀ 2600 kg/m3㸪2600 kg/m3㸪2437.5 kg/m3㸪2294.1 kg/m3 ࡛࠶ࡿ㸬௨ୗ㸪ࡇࡢࣔ
ࢹࣝࢆ࢜ࣜࢪࢼࣝࣔࢹࣝ࡜࿧ࡪࡇ࡜࡟ࡍࡿ㸬
83
Fig. 6.2
2 1D FEM m
model of a ceentrifugal com
mpressor
Table
T
6.1
D
Dimensions of
o shaft elemeents
Shaft
S
element
Length
(mm)
Outer diameeter
(mm)
1, 2, 17, 18
15.0
10.0
3-5, 14-16
25.0
12.0
6, 13
25.0
13.5
7-12
15.0
13.5
Table 6.2
Dimensio
ons of discs
Disc
Width
(mm)
Outer diameeter
(mm)
Inner diiameter
(mm
m)
1
11.0
75.0
13.5
2
11.5
75.0
13.5
3
13.0
750
13.5
4
14.0
750
13.5
≉ᛶ⾜ิ
ิࡢྠᐃ
࢜ࣜࢪࢼࣝࣔࢹ
ࢹࣝ࡟ᑐࡋ࡚
࡚㍈ࡢᐦᗘ㸪 ࣖࣥࢢ⋡㸪࣏࢔ࢯࣥẚ
ẚ࡟ࡑࢀࡒࢀ
ࢀ -10%㸪+1
10%㸪
+10%
%ࡢㄗᕪࢆᣢ
ᣢࡕ඲࡚ࡢࢹ
ࢹ࢕ࢫࢡࡢ┤
┤ᚄ㍈࿘ࡾࡢ
ࡢ័ᛶ࣮࣓ࣔࣥࢺ࡟+10%
%ࡢㄗᕪࢆᣢ
ᣢࡘࣔ
ࢹࣝࢆึᮇࣔࢹࣝ
ࣝ࡜ࡋࡓ㸬⾲
⾲ 6.3 ࡟࢜ࣜ ࢪࢼࣝࣔࢹࣝ࡜ึᮇࣔࢹ
ࢹࣝ࡟㛵ࡍࡿ
ࡿタィኚᩘ㸦㍈ࡢ
㸪ࣖࣥࢢ⋡㸪࣏࢔ࢯࣥẚ
ẚ㸪ࢹ࢕ࢫࢡ
ࢡࡢ័ᛶ࣮ࣔ
࣮࣓ࣥࢺ㸧ࡢ
ࡢ್ࢆ♧ࡍ㸬
ᐦᗘ㸪
84
Table 6.3 Design variables of original and initial model
ȡ
E
Jdisc1
Jdisc2
Jdisc3
Ȟ
(kg/m3)
(Pa)
(kgm2)
(kgm2)
(kgm2)
Original
model
Initial
model
Jdisc4
(kgm2)
7800
2.110×1011
0.30 4.56064×105 4.77992×105 5.10716×105 5.20738×105
7020
2.321×1011
0.33 5.01670×105 5.25791×105 5.61788×105 5.72812×105
࢜ࣜࢪࢼࣝࣔࢹࣝ࠿ࡽࢩ࣑࣮ࣗࣞࢺࡋࡓ㸪๛య≉ᛶ࡜ᙎᛶ≉ᛶ࡟㛵ࡍࡿㄗᕪࢆᣢࡓ࡞࠸
ᶍᨃࡢᐇ ್࡜ࢹ࢕ࢫࢡࡢ㉁㔞࡟㛵ࡍࡿ᝟ሗ࠿ࡽ㸪ᥦ᱌ᡭἲ࡟ࡼࡗ࡚ึᮇࣔࢹࣝࡢ㍈ࡢᐦ
ᗘ ȡ㸪࣏࢔ࢯࣥẚ Ȟ㸪࠾ࡼࡧࣖࣥࢢ⋡ E ࡜ࢹ࢕ࢫࢡࡢ┤ᚄ㍈࿘ࡾࡢ័ᛶ࣮࣓ࣔࣥࢺ Jdisc ࡢ᭱
㐺್ࢆồࡵ࡚ྠᐃࡋࡓ≉ᛶ⾜ิ࠿ࡽィ⟬ࡉࢀࡿ๛య≉ᛶ࡜ᙎᛶ≉ᛶࢆ㸪࢜ࣜࢪࢼࣝࣔࢹࣝ
ࡢࡶࡢ࡜ẚ㍑ࡋࡓ㸬ࡇࡇ࡛㸪3+ ndisc=7 ಶࡢᮍ▱ኚᩘ㸦ࢹ࢕ࢫࢡࡢᩘ ndisc=4㸧࡟ᑐࡋ࡚㸪ᅛ
᭷᣺ືᩘࡢᩘࢆ nf1 =3㸪ᅛ᭷࣮ࣔࢻࡢᩘࢆ nf2 =1 ࡜ࡋ࡚㸪3+ nf1+ nf2 ×n=25 ಶ㸦⠇Ⅼᩘ n=19㸧
ࡢㄗᕪ㛵ᩘࢆタᐃࡋࡓ㸬ࡲࡓ㸪㔜ࡳಀᩘ W ࡣ㸪㉁㔞 mR 㸪㔜ᚰ఩⨨ zCG㸪㔜ᚰࢆ㏻ࡾᅇ㌿㍈
࡟┤஺ࡍࡿ㍈࿘ࡾࡢ୺័ᛶ࣮࣓ࣔࣥࢺ JCG㸪ᅛ᭷᣺ືᩘ f㸪ᅛ᭷࣮ࣔࢻ ĭ ࡟ࡘ࠸࡚ࡑࢀࡒࢀ㸪
1000㸪1000㸪2000㸪1000㸪50 ࡜ࡋࡓ㸬ྠᐃࡉࢀࡓ᣺ືゎᯒࣔࢹࣝࢆྠᐃࣔࢹࣝ࡜࿧ࡪࡇ࡜
࡟ࡍࡿ㸬
࢜ࣜࢪࢼࣝࣔࢹࣝ㸪ึᮇࣔࢹࣝ㸪࠾ࡼࡧྠᐃࣔࢹࣝ࡟㛵ࡍࡿ๛య≉ᛶ࡜ᅛ᭷᣺ືᩘࢆ㸪
࢜ࣜࢪࢼࣝࣔࢹࣝ࡟ᑐࡍࡿㄗᕪ࡜ඹ࡟ࡑࢀࡒࢀ⾲ 6.4 ࡜⾲ 6.5 ࡟♧ࡍ㸬ࡲࡓ㸪࢜ࣜࢪࢼࣝࣔ
ࢹࣝ࡟ᑐࡍࡿᅛ᭷࣮ࣔࢻࡢ MAC ್㸦Balmes and Leclere, 2006㸧ࡣ㸪ึᮇࣔࢹ࡛ࣝࡣ➨ 1 ḟ
࣮ࣔࢻ࠿ࡽ㡰࡟ࡑࢀࡒࢀ 0.9998㸪0.9993㸪0.9989㸪0.9990 ࡛࠶ࡾ㸪ྠᐃࣔࢹ࡛ࣝࡣࡑࢀࡒࢀ
1.0000㸪1.0000 㸪1.0000 㸪1.0000 ࡛࠶ࡿ㸬ࡇࢀࡽࡢ⤖ᯝ࠿ࡽ㸪ྠᐃࣔࢹࣝࡢ๛య≉ᛶ࡜ᙎ
ᛶ≉ᛶࡣ࢜ࣜࢪࢼࣝࣔࢹࣝ࡜୍⮴ࡋ࡚࠸ࡿࡇ࡜ࡀศ࠿ࡿ㸬
Table 6.4 Rigid properties for centrifugal compressor
Original
Model
Initial Model
(Error)
Identified
Model
(Error)
mc
(kg)
zCGc
(m)
JCGc
(kgm2)
JpCGc
(kgm2)
0.84821
0.17593
3.574×10-3
3.859×10-4
0.81559
(-3.836%)
0.17597
(0.023%)
3.314×10-3 4.246×10-4
(-7.269%) (10.020%)
0.84821
(0.000%)
0.17593
(0.000%)
3.574×10-3
(0.000%)
85
3.859×10-4
(0.000%)
Table 6.5
Natural frequencies for centrifugal compressor rotor
1st Mode
(Hz)
Original Model
Initial Model
(Error)
Identified Model
(Error)
444.87
2nd Mode 3rd Mode
(Hz)
(Hz)
954.00
486.15
1025.5
(9.279%) (7.495%)
444.87
954.00
(0.000%) (0.000%)
4th Mode
(Hz)
1773.9
2741.9
1878.2
(5.880%)
1773.9
(0.000%)
2886.5
(5.274%)
2741.9
(0.000%)
⠇Ⅼ 2 ࡜⠇Ⅼ 18 ࡟ྠ୍ࡢ➼౯ᨭᣢ๛ᛶࢆᣢࡘ㍈ཷࢆ㓄ࡋࡓ࣮ࣟࢱ㸫㍈ཷ⣔࡟࠾࠸࡚㸪ࢪ
ࣕ࢖ࣟຠᯝࡀ⌧ࢀ࡞࠸࣮ࣟࢱࡢᅇ㌿ᩘ N # 0 rpm ࡢሙྜ࡜ࢪࣕ࢖ࣟຠᯝࡀ⌧ࢀࡿ N
18,000
rpm ࡢሙྜ࡟࠾ࡅࡿ➨ 4 ḟ࣮ࣔࢻࡲ࡛ࡢ༴㝤㏿ᗘ࣐ࢵࣉࢆ㸪࢜ࣜࢪࢼࣝࣔࢹࣝ㸪ึᮇࣔࢹࣝ㸪
࠾ࡼࡧྠᐃࣔࢹࣝ࡟ࡘ࠸࡚ẚ㍑ࡋࡓࡶࡢࢆᅗ 6.3 ࡟♧ࡍ㸬ึᮇࣔࢹࣝࡢ୙ῶ⾶ᅛ᭷᣺ືᩘࡣ㸪
㠀ᅇ㌿᫬࡜ࢪࣕ࢖ࣟస⏝ࡀാࡃ㧗㏿ᅇ㌿᫬ࡢ཮᪉࡟࠾࠸࡚㸪࢜ࣜࢪࢼࣝࣔࢹࣝ࡜᫂☜࡞ᕪ
␗ࡀㄆࡵࡽࢀࡿ㸬୍᪉㸪ྠᐃࣔࢹࣝࡢ୙ῶ⾶ᅛ᭷᣺ືᩘࡣ㸪㠀ᅇ㌿᫬ࡔࡅ࡛࡞ࡃ㧗㏿ᅇ㌿
᫬࡟࠾࠸࡚ࡶ࢜ࣜࢪࢼࣝࣔࢹࣝ࡜୍⮴ࡋ࡚࠸ࡿࡇ࡜ࡀศ࠿ࡿ㸬
86
Undamped Natural Frequency (Hz)
F: Forw
ward
B: Backkward
4th
3rd
2nd
1st
Beaaring Stiffness (N/m)
Undamped Natural Frequency (Hz)
((a) N # 0 rpm
m
4th
h-F
3rrd-F
3rd
d-B
4th-B
2nd-B
2nd-F
1sst-F
1sst-B
Bearring Stiffness (N/m)
(b)) N 18,000 rpm
r
Fig. 6.3
Critical sp
peed maps off centrifugal compressor
c
rotor – bearingg system
ࡢ㍈ཷࢆ㓄ࡋ
ࡋ㸪ࢹ࢕ࢫࢡ
ࢡ 1 ࡟ 10 gmm/ 0°㸪ࢹ࢕ࢫࢡ 4 ࡟ 100 gmm/ 90°ࡢ୙㔮
ࡢ
⾲ 6.6 ࡢ௙ᵝࡢ
ྜࡢ㍈ཷ఩⨨
⨨࡛ࡢ୙㔮ྜ
ྜ࠸ᛂ⟅᣺ᖜ
ᖜࢆᅗ 6.4 ࡟♧ࡍ㸬ึᮇࣔ
ࣔࢹࣝࡢ୙㔮
㔮ྜ࠸
ྜ࠸ࢆ୚࠼ࡓሙྜ
᣺ᖜࡣ࢜ࣜࢪ
ࢪࢼࣝࣔࢹࣝ
ࣝ࡜᫂☜࡞ᕪ
ᕪ␗ࡀㄆࡵࡽ
ࡽࢀࡿࡢ࡟ᑐ
ᑐࡋ࡚㸪ྠᐃ
ᐃࣔࢹࣝࡢ୙
୙㔮ྜ
ᛂ⟅᣺
࠸ᛂ⟅
⟅᣺ᖜࡣ࢜ࣜ
ࣜࢪࢼࣝࣔࢹ
ࢹࣝ࡜୍⮴ࡋ
ࡋ࡚࠸ࡿ㸬
87
Table 6.6 Bearing
B
param
meters for cen
ntrifugal com
mpressor rotorr
Bearingg
Node
#1
#2
2
18
kxx
kyy
cyx
kxy
kyx
cxx
cxy
cyy
N/m) (N/m
m) (N/m) (N/m)
(
(Ns/m
m) (Ns/m) (N
Ns/m) (Ns/m
m)
(N
6
6
6
6
3
3
3
2.0×10 1.5×100 1.0×10 3.0×10 2.0×10
0 1.5×10 1..0×10 3.0×1
103
6
6
6
6
3
3
3
2.0×10 1.5×100 1.0×10 3.0×10 2.0×10
0 1.5×10 1..0×10 3.0×1
103
Magnitude (ȝm0-P)
Rootor Spin Speed (rpm)
Magnitude (ȝm0-P)
((a) Bearing #1
#
Rottor Spin Speed
d (rpm)
((b) Bearing #2
#
Fig. 6.4
Unnbalance response at bearinngs for centriifugal compreessor rotor – bbearing systeem
ࡢ㍈ཷࢆ㓄ࡋ
ࡋࡓሙྜࡢ㸪
66,000rpm ࠿ࡽ
ࡽ 90,000rpm
m ࡲ࡛ࡢᅇ㌿
㌿ᩘ⠊ᅖ࡟࠾ࡅࡿ㸪
⾲ 6.6 ࡢ௙ᵝࡢ
ḟ࠿ࡽ 2 ␒┠ࡢ」⣲ᅛ
␒
ᅛ᭷್ࢆ㸪࢜
࢜ࣜࢪࢼࣝࣔࢹ
ࢹࣝ㸪ึᮇࣔ
ࣔࢹࣝ㸪࠾ࡼ
ࡼࡧྠᐃࣔࢹ
ࢹࣝ࡟
᭱పḟ
ࡘ࠸࡚
࡚ẚ㍑ࡋࡓࡶ
ࡶࡢࢆᅗ 6.5
5 ࡟♧ࡍ㸬ึ ᮇࣔࢹࣝࡢ」
」⣲ᅛ᭷್ࡣ
ࡣ࢜ࣜࢪࢼࣝ
ࣝࣔࢹࣝ࡜᫂
᫂☜࡞
ᕪ␗ࡀ
ࡀㄆࡵࡽࢀࡿࡢ࡟ᑐࡋ࡚
࡚㸪ྠᐃࣔࢹ
ࢹࣝࡢ」⣲ᅛ
ᅛ᭷್ࡣ࢜ࣜ
ࣜࢪࢼࣝࣔࢹ
ࢹࣝ࡜୍⮴ࡋ
ࡋ࡚࠸
ࡿࡇ࡜ࡀศ࠿ࡿ㸬≉࡟㸪Ᏻᐃ
ᐃᛶࢆホ౯ࡍ ࡿୖ࡛㔜せ࡞」⣲ᅛ᭷್
್ࡢᐇ㒊ࡀ㈇
㈇࠿ࡽṇ࡜࡞
࡞ࡿ㸪
ṇ࠿ࡽ㈇࡜࡞
࡞ࡿᅇ㌿ᩘࡣ
ࡣ㸪ึᮇࣔࢹ
ࢹ࡛ࣝࡣ࢜ࣜ
ࣜࢪࢼࣝࣔࢹ
ࢹࣝ࡜ྠᐃࣔ
ࣔࢹࣝ
ࡍ࡞ࢃ
ࢃࡕῶ⾶ࡀṇ
ࡢ⣙ 72,000rpm ࡼࡾ⣙
ࡼ
6,000
0rpm 㧗࠸⣙ 78,000rpm ࡜࡞ࡗ࡚࠾ࡾ㸪ึᮇࣔࢹ
ࢹࣝࡢㄗᕪࡀ
ࡀᏳᐃ
ࡢண ⢭ᗘ࡟
࡟ཬࡰࡍᙳ㡪
㡪ࡀ↓ど࡛ࡁ
ࡁ࡞࠸ࡇ࡜ࡀ
ࡀศ࠿ࡿ㸬
㝈⏺ࡢ
88
Imaginary part (rad/s)
Real part (raad/s)
Fiig. 6.5
2nd complex eigeenvalues for centrifugal
c
co
ompressor rottor
⤖ゝ
୍⯡
⯡࡟㸪ᐇ㦂࡟
࡟ࡼࡗ࡚ᮦᩱ
ᩱ≉ᛶ್࡟ㄗ
ㄗᕪࢆྵࢇࡔ
ࡔ࣮ࣟࢱࡢ᣺
᣺ືゎᯒࣔࢹ
ࢹࣝࢆ㧗⢭ᗘ
ᗘ໬ࡍ
ࡿሙྜ
ྜ㸪㠀ᅇ㌿㸪⮬⏤ᨭᣢ≧
≧ែࡢ࣮ࣟࢱ
ࢱࡢ᣺ືヨ㦂
㦂࡛ᚓࡽࢀࡿᙎᛶ≉ᛶ㸦࿘
࿘Ἴᩘᛂ⟅㛵
㛵ᩘ㸪
ᅛ᭷᣺
᣺ືᩘ㸪ᅛ᭷
᭷࣮ࣔࢻ㸧࡟
࡟ᇶ࡙࠸࡚㸪 ᮦᩱ≉ᛶ್
್ࢆ᭱㐺໬ࡍ
ࡍࡿ᪉ἲࡀ᥇
᥇ࡽࢀ࡚࠸ࡿ
ࡿ㸬ࡇ
ࢀ࡟ᑐ
ᑐࡋ࡚㸪ᙎᛶ
ᛶ≉ᛶ㸦ᅛ᭷
᭷᣺ືᩘ㸪ᅛ
ᅛ᭷࣮ࣔࢻ㸧࡟ຍ࠼࡚๛
๛య≉ᛶ㸦㉁
㉁㔞㸪㔜ᚰ఩
఩⨨㸪
័ᛶࣔ
࣮࣓ࣔࣥࢺ㸧
㸧ࡢᐇ ್࡟
࡟ᇶ࡙࠸࡚㸪 ඲࡚ࡢᮦᩱ
ᩱ≉ᛶ್ࢆྠ
ྠ᫬࡟᭱㐺໬
໬ࡍࡿ᪉ἲࢆ
ࢆᥦ᱌
ࡋࡓ㸬
㸬
㍈ࡢ
ࡢྛ≉ᛶ⾜ิ
ิࢆ㸪᪂ࡓ࡟
࡟ᑟධࡋࡓ㉁
㉁㔞ಀᩘ࡜័
័ᛶಀᩘ㸪ࡏ
ࡏࢇ᩿ಀᩘ㸪 ᭤ࡆ๛ᛶ࡞
࡞࡝࡛
ᐃ⩏ࡋ㸪ࢹ࢕ࢫࢡ
ࢡࡢ≉ᛶ⾜ิ
ิࢆ㉁㔞㸪┤
┤ᚄ㍈࿘ࡾࡢ
ࡢ័ᛶ࣮࣓ࣔ
࣓ࣥࢺ㸪ཌࡉ
ࡉ࡛ᐃ⩏ࡍࡿ
ࡿࡇ࡜
ࡗ࡚㸪ྛ≉ᛶ
ᛶ⾜ิࡢせ⣲
⣲ࡢ≀⌮ⓗ࡞
࡞ព࿡ࡢ᫂☜
☜໬ࢆᅗࡗࡓ
ࡓ㸬
࡟ࡼࡗ
4ẁ
ẁ㐲ᚰᅽ⦰ᶵ
ᶵ࣮ࣟࢱࢆྲྀ
ྲྀࡾୖࡆ࡚㸪ㄗᕪࡢ↓࠸ᶍ
ᶍᨃࡢ๛య≉
≉ᛶ࡜ᙎᛶ≉
≉ᛶ࡟㛵ࡍࡿ
ࡿ᝟ሗ
࠿ࡽ㸪
㸪ᐇ㝿࡟඲࡚
࡚ࡢ≉ᛶ⾜ิ
ิࡢྠᐃࢆ⾜
⾜ࡗࡓ㸬ྠᐃ
ᐃ⢭ᗘ࡟ࡘ࠸
࠸࡚ࡣ㸪௨ୗ
ୗࡢࡶࡢࢆㄗ
ㄗᕪࡢ
↓࠸᣺
᣺ືゎᯒࣔࢹ
ࢹࣝࡢ್࡜ẚ
ẚ㍑ࡍࡿࡇ࡜
࡜࡟ࡼࡗ᳨࡚
᳨ドࡋࡓ㸬
࣭ ࣮ࣟࢱࡢ඲
඲㉁㔞㸪㔜ᚰ఩
఩⨨㸪㔜ᚰࢆ
ࢆ㏻ࡾᅇ㌿㍈
㍈࡟┤஺ࡍࡿ
ࡿ㍈࿘ࡾࡢ୺
୺័ᛶ࣮࣓ࣔࣥࢺ㸪
ᴟ័ᛶ࣮ࣔ
࣮࣓ࣥࢺ
ࡢ࣮ࣟࢱࡢᅛ
ᅛ᭷᣺ືᩘ࡜ ᅛ᭷࣮ࣔࢻ
࣭ 㠀ᅇ㌿᫬ࡢ
⾶≉ᛶࡀ᪤▱
▱ࡢ㍈ཷ࡛ᨭ
ᨭᣢࡉࢀࡓ࣮ࣟࢱࡢ㸪ᴟప
ప㏿ᅇ㌿᫬࡜
࡜㧗㏿ᅇ㌿᫬
᫬࡟࠾
࣭ ๛ᛶ࡜ῶ⾶
ࡅࡿ༴㝤㏿
㏿ᗘ࣐ࢵࣉ㸪୙㔮ྜ࠸ᛂ
ᛂ⟅㸪」⣲ᅛ᭷್
ࡢ⤖ᯝ㸪ྠᐃ
ᐃㄗᕪࢆ↓ど
ど࡛ࡁࡿ࡯࡝
࡝㧗࠸⢭ᗘ࡛
࡛≉ᛶ⾜ิࢆ
ࢆྠᐃࡍࡿࡇ
ࡇ࡜ࡀ࡛ࡁࡿ
ࡿࡇ࡜
ࡑࡢ
ࢆ☜ㄆ
ㄆࡋࡓ㸬
89
≉ᚩࢆ௨ୗ࡟ࡲ࡜ࡵࡿ㸬
࣭ ᣺ືヨ㦂ࡣ㸪㠀ᅇ㌿㸪⮬⏤ᨭᣢ≧ែ࡛ᐇ᪋ࡍࢀࡤࡼ࠸㸬
࣭ ᅇ㌿㏿ᗘ౫Ꮡ㡯ࢆྵࡴ඲࡚ࡢ≉ᛶ⾜ิࢆྠᐃྍ⬟࡛࠶ࡿ㸬
࣭ ᙎᛶ≉ᛶࡔࡅ࡛࡞ࡃ๛య≉ᛶࡶ฼⏝ࡍࡿࡓࡵ㸪㧗⢭ᗘ࡞≉ᛶ⾜ิࡀྠᐃࡉࢀࡿ㸬
࣭ 㔜ࡳಀᩘ࡟ࡼࡗ࡚ィ ㄗᕪࢆ⪃៖ࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
࣭ 㔜ࡳಀᩘ࡜チᐜㄗᕪࢆタᐃࡍࢀࡤ㸪⮬ື࡛」ᩘࡢᮍ▱ࣃ࣓࣮ࣛࢱࡀྠ᫬࡟᭱㐺໬ࡉࢀ
ࡿࡓࡵ㸪᭱㐺໬ࡍࡿࣃ࣓࣮ࣛࢱࡢ㑅ᐃࡀ୙せ࡛࠶ࡿ㸬
ཧ⪃ᩥ⊩
Balmes, E. and Leclere, J. M., Structural Dynamics Toolbox / FEMLink user’s guide version 5.3,
SDTools (2006), pp.416 – 417.
Coleman, T.F. and Li, Y., An Interior, Trust Region Approach for Nonlinear Minimization Subject
to Bounds, SIAM Journal on Optimization, Vol. 6, No. 2 (1996), pp. 418–445.
Fareed, A.M. and Wahl, F., Identification of Rigid Body Parameters Using Experimental Modal
Analysis Data, Technische Mechanik, Band 21, Heft 1 (2001), pp. 63-75.
Lagarias, J. C., Reeds, J. A., Wright, M. H., and Wright, P. E., Convergence properties of the
nelder-mead simplex method in low dimensions, SIAM Journal of Optimization, Vol. 9, No. 1
(1998), pp. 112-147.
㛗Ụ ಙ㢧, ᚋ⸨ ▱ఙ, ࣮ࣟࢱࡢ᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ≉ᛶ⾜ิࡢᐇ㦂ྠᐃ㸦⌮ㄽ࡜ࢩ࣑ࣗࣞ
࣮ࢩࣙࣥ㸧, ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟 C ⦅, Vol.79, No.803 (2013), pp.2314-2328.
Okuma, M. and Heylen, W., MCK ྠᐃἲ࡟ࡼࡿ 3 ḟඖᵓ㐀≀ࡢ๛య≉ᛶࡢྠᐃ, ᪥ᮏᶵᲔᏛ
఍ D&D2000 ㅮ₇ㄽᩥ㞟, No. 301 (2000).
90
➨ ❶ ⥲ᣓ
ᮏ◊✲࡟ࡼࡾᚓࡽࢀࡓᡂᯝ
1 ḟඖᱱࡢ᭷㝈せ⣲ἲࡀ㐺⏝ࡉࢀࡓ࣮ࣟࢱࡢ㍈᣺ືゎᯒࣔࢹࣝ࡟࠾࠸࡚㸪᭤ࡆ᣺ືゎᯒࣔ
ࢹࣝࡢ㧗⢭ᗘ໬࡟ᐤ୚ࡍࡿ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡋࡓࣔࢹࣝ໬᪉ἲ㸪ᔐࡵ
ྜ࠸㒊ࡢࣔࢹࣝ໬᪉ἲ㸪࠾ࡼࡧ᣺ືヨ㦂⤖ᯝ࡟ᇶ࡙࠸࡚᣺ືゎᯒࣔࢹࣝࢆ㧗⢭ᗘ໬ࡍࡿ᪉
ἲ࡜㸪ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬࡟ᐤ୚ࡍࡿ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖
ࡋࡓࣔࢹࣝ໬᪉ἲࢆ☜❧ࡋࡓ㸬
᭤ࡆ᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬࡛ࡣ㸪㍈┤ᚄኚ໬㒊ࡢᑠᚄ㍈ഃ࡟タࡅࡓ㛗ࡉ 9d/32㸦d ࡣ
ᑠᚄ㍈ࡢ┤ᚄ㸧ࡢせ⣲ࡢ᭤ࡆ๛ᛶࢆ㐺ษ࡟ᙅࡵࡿࡇ࡜࡟ࡼࡗ࡚㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶ
ኚᙧ࡟ࡼࡿࡳ࠿ࡅࡢ㍈ࡢ᭤ࡆ๛ᛶపୗࢆ⾲⌧ࡍࡿ᪉ἲࢆ㛤Ⓨࡋ㸪3 ḟඖゎᯒࣔࢹࣝ࡜ᐇ㦂࡟
ࡼࡗ࡚㧗⢭ᗘ໬ࡢຠᯝࢆ☜ㄆࡋࡓ㸬ࡲࡓ㸪ᔐࡵྜ࠸㒊ࡢ➼౯࡞᭤ࡆ๛ᛶࢆ୚࠼ࡿ㍈ࡢእᚄ
ࡀ㸪ᔐࡵྜࢃࡏࡿࢫ࣮ࣜࣈࡢእᚄ࡜࡯ࡰ୍⮴ࡍࡿࡇ࡜ࢆ㸪ᔐࡵྜ࠸㒊ࡢᦶ᧿ࢆ⪃៖ࡋࡓ㠀
⥺ᙧ㟼ゎᯒ᥋ゐ࡜ᐇ㦂࡟ࡼࡗ࡚♧ࡋࡓ㸬ࡉࡽ࡟㸪᣺ືヨ㦂⤖ᯝ࡟ᇶ࡙࠸࡚᣺ືゎᯒࣔࢹࣝ
ࢆ㧗⢭ᗘ໬ࡍࡿ᪉ἲ࡜ࡋ࡚ࡣ㸪㉁㔞㸪㔜ᚰ఩⨨㸪័ᛶ࣮࣓ࣔࣥࢺ㸪ᅛ᭷᣺ືᩘ㸪࠾ࡼࡧᅛ
᭷࣮ࣔࢻ࡟㛵ࡍࡿ᝟ሗࢆཧ↷ࢹ࣮ࢱ࡜ࡋ࡚㸪ㄗᕪࢆྵࢇࡔ඲࡚ࡢᮦᩱ≉ᛶ್ࢆྠ᫬࡟㧗⢭
ᗘ໬ࡋ㸪᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ඲࡚ࡢ≉ᛶ⾜ิ㸦㉁㔞⾜ิ㸪ῶ⾶⾜ิ㸪๛ᛶ⾜ิ㸪࠾ࡼࡧࢪ
ࣕ࢖ࣟ⾜ิ㸧ࢆྠᐃࡍࡿ᪉ἲࢆ㛤Ⓨࡋ㸪ཧ↷ࢹ࣮ࢱ࡟ㄗᕪࡀ↓࠸ሙྜ࡟ࡣ㸪ㄗᕪࡀ↓ど࡛
ࡁࡿ࡯࡝㧗⢭ᗘ࡟≉ᛶ⾜ิࢆྠᐃ࡛ࡁࡿࡇ࡜ࢆ♧ࡋࡓ㸬
ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬࡛ࡣ㸪㍈┤ᚄኚ໬㒊ࡢᑠᚄ㍈ഃ࡟タࡅࡓ㛗ࡉ
3ʌd/{64(1+0.3382ȕ3+0.0815ȕ5+0.1144ȕ6+0.0125ȕ7)}㸦ȕ ࡣᑠᚄ㍈࡜኱ᚄ㍈ࡢ┤ᚄẚ㸪ȕ<1㸧ࡢせ
⣲ࡢࡡࡌࡾ๛ᛶࢆ㐺ษ࡟ᙅࡵࡿࡇ࡜࡟ࡼࡗ࡚㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧ࡟ࡼࡿࡳ࠿ࡅ
ࡢ㍈ࡢࡡࡌࡾ๛ᛶపୗࢆ⾲⌧ࡍࡿ᪉ἲࢆ㛤Ⓨࡋ㸪3 ḟඖゎᯒࣔࢹࣝ࡟ࡼࡗ࡚㧗⢭ᗘ໬ࡢຠᯝ
ࢆ☜ㄆࡋࡓ㸬
᭤ࡆ᣺ືゎᯒࣔࢹࣝ࡜ࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢࡑࢀࡒࢀ࡟࠾࠸࡚㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢ
ᙎᛶኚᙧࢆ⪃៖ࡍࡿ᪉ἲࡣ㸪ᗈࡃᬑཬࡋ࡚࠸ࡿ 1 ḟඖᱱࡢ㍈᣺ືゎᯒ⎔ቃ࡟㸪ィ⟬㈇Ⲵࢆ
≛≅࡟ࡍࡿࡇ࡜࡞ࡃᐜ᫆࡟㐺⏝࡛ࡁ㸪ᅇ㌿ᶵᲔࡢタィࡸ᣺ືၥ㢟ࡢᑐ⟇᳨ウ࡛㔜せ࡞ᅛ᭷
᣺ືᩘࡢண ⢭ᗘࢆ㧗ࡵࡿࡇ࡜ࡀ࡛ࡁࡿ㸬ࡲࡓ㸪ලయ౛࡜ࡋ࡚ᐇᶵ┦ᙜࡢࢱ࣮ࣅࣥ㸫Ⓨ㟁
ᶵ࣮ࣟࢱࢆྲྀࡾୖࡆ࡚㸪ࡑࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝ࡟㛤Ⓨࡋࡓ᪉ἲࢆ㐺⏝ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪
పḟ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡀᚑ᮶ἲ࡟ᑐࡋ࡚᭱኱࡛⣙ 5%పୗࡍࡿࡇ࡜ࢆ☜ㄆࡋ㸪ࡡࡌࡾ᣺ື
ゎᯒࣔࢹࣝ࡟㛤Ⓨࡋࡓ᪉ἲࢆ㐺⏝ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪పḟ࣮ࣔࢻࡢᅛ᭷᣺ືᩘࡀᚑ᮶ἲ࡟
ᑐࡋ࡚᭱኱࡛⣙ 2%పୗࡍࡿࡇ࡜ࢆ☜ㄆࡋࡓ㸬
㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡍࡿ᪉ἲ࡜ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࢆࢫ࣮ࣜࣈእᚄ࡜
ࡍࡿ᪉ἲࢆ㸪1 ḟඖᱱࡢ㍈᣺ືゎᯒ⎔ቃ࡟㐺⏝ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪ィ⟬㈇Ⲵࡀᑠࡉࡃ㧗⢭ᗘ
࡞᭤ࡆ᣺ືゎᯒ࡜ࡡࡌࡾ᣺ືゎᯒࢆᐇ⌧࡛ࡁࡿ㸬ࡲࡓ㸪㉁㔞㸪㔜ᚰ఩⨨㸪័ᛶ࣮࣓ࣔࣥࢺ㸪
ᅛ᭷᣺ືᩘ㸪࠾ࡼࡧᅛ᭷࣮ࣔࢻ࡟㛵ࡍࡿ᝟ሗ࠿ࡽ᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ඲࡚ࡢ≉ᛶ⾜ิࢆྠ
ᐃࡍࡿ᪉ἲࡣ㸪ᐇ㦂࣮ࣔࢻゎᯒ⤖ᯝ࡟ᇶ࡙࠸ࡓ࣮ࣟࢱࡢ᣺ືゎᯒࣔࢹࣝࡢ㧗⢭ᗘ໬ࢆᐇ⌧
91
ࡍࡿࡓࡵ࡟᭷ຠ࡟ά⏝࡛ࡁࡿ㸬
௒ᚋࡢᒎᮃ
ᅇ㌿ᶵᲔࡢ᣺ືゎᯒࡢ᪉ἲ࡜ࡋ࡚⌧ᅾᬑཬࡋࡘࡘ࠶ࡿ㸪3 ḟඖࡢࢯࣜࢵࢻせ⣲࡛ヲ⣽࡟ࣔ
ࢹࣝ໬ࡍࡿ 3D ࣔࢹࣝゎᯒࡀ㸪௒ᚋࡢࢥࣥࣆ࣮ࣗࢱࡢᛶ⬟ୖ᪼࡟కࡗ࡚㸪ᬑཬࡢᗘྜ࠸ࢆቑ
ࡍࡶࡢ࡜᥎ ࡉࢀࡿ㸬ࡋ࠿ࡋ㸪ࡼࡾ⡆౽࡞ 1 ḟඖᱱࣔࢹࣝ࡟ࡼࡿゎᯒࡶ㸪1.1 ⠇࡛㏙࡭ࡓ⧞
ࡾ㏉ࡋࡢ」⣲ᅛ᭷್ゎᯒࢆ⾜࠺ᚲせࡢ࠶ࡿሙྜ࡞࡝࡛ࡣ᫬㛫ࢥࢫࢺ㠃࠿ࡽ᭷฼࡛࠶ࡿ≧ἣ
ࡣᙜศࡢ㛫ࡣኚࢃࡽ࡞࠸࡜ᛮࢃࢀࡿ㸬ࡋࡓࡀࡗ࡚㸪1 ḟඖᱱࣔࢹࣝ࡟ࡼࡿィ⟬⢭ᗘࢆ㧗ࡵࡿ
ࡇ࡜ࡣ௒ᚋࡶ᭷ຠ࡜ุ᩿ࡉࢀࡿ㸬
➨ 3 ❶࡜➨ 5 ❶࡛♧ࡋࡓ㸪㍈┤ᚄኚ໬㒊᩿㠃ࡢᙎᛶኚᙧࢆ⪃៖ࡍࡿ᪉ἲࡢᇶ♏࡜࡞ࡿ 3
ḟඖᙎᛶㄽࡣ㸪1940 ᖺ௦࠿ࡽ 1960 ᖺ௦ࡲ࡛࡟࡯ࡰ᏶ᡂࡉࢀࡓ⌮ㄽ࡛࠶ࡾ㸪ቃ⏺᮲௳࡜೫ᚤ
ศ᪉⛬ᘧ࡛⾲ࡉࢀࡿ᪤▱ࡢᛂຊ㸪ࡦࡎࡳ㸪ኚ఩ࡢ㛵ಀᘧ࠿ࡽ㸪௵ពࡢᗙᶆ఩⨨ࡢᛂຊ㸪ࡦ
ࡎࡳ㸪ኚ఩ࡢཝᐦゎࢆᚓࡿࡇ࡜ࡀ࡛ࡁࡿࡀ㸪ᙧ≧ࡸቃ⏺᮲௳ࡀ」㞧࡟࡞ࡿ࡜ᢅ࠸ࡀⴭࡋࡃ
㞴ࡋࡃ࡞ࡿ㸬ࡑࡢࡓࡵ㸪3 ḟඖࣔࢹࣝࡢ᭷㝈せ⣲ゎᯒࡀᬑཬࡋ࡚㸪ᙧ≧࡜ᮦᩱ≉ᛶࢆᐃ⩏ࡍ
ࢀࡤᩘ್ゎࢆᚓࡿࡇ࡜ࡀ࡛ࡁࡿ⌧ᅾ࡛ࡣ㸪ᕤᏛศ㔝࡛ 3 ḟඖᙎᛶㄽࡑࡢࡶࡢࢆᢅ࠺ᶵ఍ࡣ
ᑡ࡞ࡃ࡞ࡗ࡚࠸ࡿ㸬ᮏ◊✲࡛ࡣ㸪ࡇࡢ 3 ḟඖᙎᛶㄽ࠿ࡽᑟ࠿ࢀࡿ⤖ᯝࢆ 1 ḟඖᱱࣔࢹࣝ࡟
㐺⏝ࡍࡿࡇ࡜࡟ࡼࡗ࡚ゎᯒ⢭ᗘࢆ㧗ࡵࡿࡇ࡜ࡀ࡛ࡁࡿࡇ࡜ࢆ♧ࡋࡓ㸬
ࡲࡓ㸪5.2 ⠇࡛♧ࡋࡓ㸪
3 ḟඖᙎᛶㄽ࠿ࡽᑟ࠿ࢀࡓࢺࣝࢡ T ࡜๛య㍈ࡢᅇ㌿ゅ ș ࡢ㛵ಀᘧ㸦5.4㸧ࡣ㸪ࡇࢀࡲ࡛࡟ࣟ
࣮ࢱࡢࡡࡌࡾ᣺ືゎᯒ࡟㐺⏝ࡉࢀ࡚࠸࡞࠸ࡶࡢ࡛࠶ࡿ㸬ࡇࡢ౛ࡀ♧ࡍᵝ࡟㸪࣮ࣟࢱࢲ࢖ࢼ
࣑ࢡࢫࡢศ㔝࡛᥇⏝ࡉࢀ࡚࠸࡞࠸㸪᭷⏝࡞᪤Ꮡ◊✲ࡀᏑᅾࡍࡿྍ⬟ᛶࡀ࠶ࡿࡓࡵ㸪௒ᚋࡶ
ᖜᗈ࠸ᖺ௦࡜ศ㔝ࡢ▱ぢࢆ᭷ຠά⏝࡛ࡁࡿࡼ࠺ど㔝ࢆᗈࡆࡓ◊✲άືࢆ㐍ࡵࡿࡘࡶࡾ࡛࠶
ࡿ㸬
ࡲࡓ㸪➨ 6 ❶࡛♧ࡋࡓ㸪㉁㔞㸪㔜ᚰ఩⨨㸪័ᛶ࣮࣓ࣔࣥࢺ㸪ᅛ᭷᣺ືᩘ㸪࠾ࡼࡧᅛ᭷ࣔ
࣮ࢻࢆධຊࣃ࣓࣮ࣛࢱ࡜ࡋ࡚᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ඲࡚ࡢ≉ᛶ⾜ิࢆྠᐃࡍࡿ᪉ἲࢆ㸪ᐇ㝿
ࡢ࣮ࣟࢱ࡟㐺⏝ࡍࡿ࡟ࡣ㸪ධຊࣃ࣓࣮ࣛࢱࡢ⢭ᗘࢆ⪃៖ࡋ࡚㔜ࡳಀᩘࢆ㐺ษ࡞್࡟タᐃࡍ
ࡿᚲせࡀ࠶ࡿ㸬ࡇࡢⅬࢆ௒ᚋࡢㄢ㢟࡜ࡋ࡚ᤊ࠼㸪⡆౽࡟౑⏝࡛ࡁࡿࡶࡢ࡟ࡍࡿࡓࡵࡢ◊✲
ࢆ㐍ࡵࡿࡘࡶࡾ࡛࠶ࡿ㸬
92
ㅰ㎡
ᮏ◊✲ࡣ㸪ⴭ⪅ࡀ㫽ྲྀ኱Ꮫ኱Ꮫ㝔ᕤᏛ◊✲⛉༤ኈᚋᮇㄢ⛬ᅾᏛ୰࡟㸪ྠ኱Ꮫ㝔ᶵᲔᏱᐂ
ᕤᏛᑓᨷ ᚋ⸨▱ఙᩍᤵࡢᣦᑟࡢࡶ࡜࡟⾜ࡗࡓࡶࡢ࡛࠶ࡿ㸬◊✲ࡢ㐙⾜࡟࠶ࡓࡾ㸪ᚋ⸨ᩍᤵ
࠿ࡽࡣᖜᗈ࠸どⅬ࠿ࡽ᠓ษ୎ᑀ࡞ᚚᣦᑟࢆ㈷ࡾࡲࡋࡓ㸬ㅽࢇ࡛ឤㅰࡢពࢆ⾲ࡋࡲࡍ㸬Ꮫ఩
ㄽᩥసᡂ࡟࠶ࡓࡾ㸪㈗㔜࡞ᚚຓゝࢆ㡬ࡁࡲࡋࡓ㸪ྠ኱Ꮫ㝔♫఍ᇶ┙ᕤᏛᑓᨷ ㇂ཱྀ᭸௦ᩍᤵ
࡜ྠ኱Ꮫ㝔ᶵᲔᏱᐂᕤᏛᑓᨷ ᑠฟ㝯ኵᩍᤵ࡟ឤㅰࡢពࢆ⾲ࡋࡲࡍ㸬ࡲࡓ㸪ᒚಟ⛉┠࡟࡚ࡈ
ᣦᑟ㡬ࡁࡲࡋࡓ㸪ྠ኱Ꮫ㝔ᶵᲔᏱᐂᕤᏛᑓᨷ すᮧṇ἞ᩍᤵ࡜ᨾ ᐑ㏆ᖾ㐓ᩍᤵ࡟ឤㅰࡢព
ࢆ⾲ࡋࡲࡍ㸬
ᮏ◊✲ࡢᶵ఍ࢆ୚࠼࡚㡬ࡁࡲࡋࡓ㸪ඖ ᕝ㔜ࢸࢡࣀࣟࢪ࣮ᰴᘧ఍♫ ᑠᯘ⚽࿴⌮஦㸦⌧ ⚽
࿴ᛂ⌮஦ົᡤ௦⾲㸧㸪ඖ ᕝ㔜ࢸࢡࣀࣟࢪ࣮ᰴᘧ఍♫ ྜྷ⏣⚽๎ྲྀ⥾ᙺ㸪ඖ ᕝ㔜ࢸࢡࣀࣟࢪ
࣮ᰴᘧ఍♫ ோ⎼ᐶኴ㒊㛗㸦⌧ ᕝ㔜ࢸࢡࣀࣟࢪ࣮ᰴᘧ఍♫ ୺ᖿ㸧࡟ឤㅰࡢពࢆ⾲ࡋࡲࡍ㸬
≉࡟㸪ඖࡢୖྖ࡛࠶ࡿᑠᯘ⚽࿴ᵝ࡟ࡣ㸪ከ኱࡞ࡿࡈᣦᑟࢆ㡬ࡁࡲࡋࡓ㸬ࡇࡇ࡟ᨵࡵ࡚ឤㅰ
⏦ࡋୖࡆࡲࡍ㸬ᒸᒣ኱Ꮫ⏘Ꮫᐁ⼥ྜࢭࣥࢱ࣮ ྂụ἞Ꮥᐈဨᩍᤵ࠿ࡽࡣ㸪ᅾᏛ୰࠾఍࠸ࡍࡿ
ᗘ࡟⃭ບࡢ࠾ゝⴥࢆ㡬ࡁ㸪኱ኚບࡳ࡟࡞ࡾࡲࡋࡓ㸬ࡇࡇ࡟ᨵࡵ࡚ឤㅰ⏦ࡋୖࡆࡲࡍ㸬
ࡲࡓ㸪ᔐࡵྜ࠸㒊ࡢ᭤ࡆ๛ᛶࡢ᳨ウ࡟࠶ࡓࡾ㸪ゎᯒ㠃࡛༠ຊࡋ࡚㡬ࡁࡲࡋࡓ㸪ᕝ㔜ࢸࢡ
ࣀࣟࢪ࣮ᰴᘧ఍♫ タィࢯ࣮ࣜࣗࢩࣙࣥ㒊ࡢ⋢ᮌ฼⿱㒊㛗㸪ᒣᆅᡂ୍୺ᖿ㸪ᮧᒣ▱❶୺ᖿ㸪
బᒸಇḟ㑻୺ᰝ࡟ឤㅰࡢពࢆ⾲ࡋࡲࡍ㸬≉࡟ゎᯒࢆᐇ⾜ࡋ࡚㡬ࡁࡲࡋࡓᮧᒣ▱❶୺ᖿ࡟ࡣ㸪
῝ࡃឤㅰ⏦ࡋୖࡆࡲࡍ㸬◊✲ࡢ㐙⾜࡟࠶ࡓࡾ㸪ᗄᗘ࡜࡞ࡃࢫࢣࢪ࣮ࣗࣝ㠃࡛༠ຊࡋ࡚㡬࠸
ࡓᡤᒓ㒊㛛࡛࠶ࡿ᣺ືᢏ⾡ㄢࡢ᪉ࠎ࡟ឤㅰࡢពࢆ⾲ࡋࡲࡍ㸬
93
㛵㐃ሗ࿌
࠙➨ 3 ❶ࠚ
ㄽᩥ
㢟┠ ࣮ࣟࢱࡢ 1 ḟඖᱱࣔࢹࣝࡢィ⟬⢭ᗘྥୖࡢࡓࡵࡢ㍈┤ᚄኚ໬㒊ࡢࣔࢹࣝ໬
ⴭ⪅ྡ 㛗Ụ ಙ㢧㸪ᚋ⸨ ▱ఙ
Ꮫ⾡㞧ㄅྡ㸦ᕳ㸪ྕ㸪㡫㸧 ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟㸦80㸪816㸪DR0237㸧
Ⓨ⾜ᖺ᭶᪥ ᖹᡂ 26 ᖺ 8 ᭶ 25 ᪥
࠙➨ 4 ❶ࠚ
Ꮫ⾡ㅮ₇
㢟┠ ᔐྜ㒊ࢆᣢࡘ࣮ࣟࢱࡢ๛ᛶ⾜ิࡢྠᐃ
ⴭ⪅ྡ 㛗Ụ ಙ㢧㸪ᚋ⸨ ▱ఙ
Ꮫ⾡㞧ㄅྡ Dynamics and Design Conference 2013
Ⓨ⾜ᖺ᭶᪥ ᖹᡂ 25 ᖺ 8 ᭶ 29 ᪥
࠙➨ 6 ❶ࠚ
ㄽᩥ
㢟┠ ࣮ࣟࢱࡢ᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ≉ᛶ⾜ิࡢᐇ㦂ྠᐃ㸦⌮ㄽ࡜ࢩ࣑࣮ࣗࣞࢩࣙࣥ㸧 ⴭ⪅ྡ 㛗Ụ ಙ㢧㸪ᚋ⸨ ▱ఙ Ꮫ⾡㞧ㄅྡ㸦ᕳ㸪ྕ㸪㡫㸧 ᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟㸦C ⦅㸧㸦79㸪803㸪2314㹼2328㸧
Ⓨ⾜ᖺ᭶᪥ ᖹᡂ 25 ᖺ 7 ᭶ 25 ᪥
Ꮫ⾡ㅮ₇
㢟┠ ᐇ㦂࡟ࡼࡿࢹ࢕ࢫࢡ௜ࡁ࣮ࣟࢱࡢ᣺ື᪉⛬ᘧ࡟࠾ࡅࡿ≉ᛶ⾜ิࡢྠᐃ
㸦⌮ㄽ࡜ࢩ࣑࣮ࣗࣞࢩࣙࣥ㸧
ⴭ⪅ྡ 㛗Ụ ಙ㢧㸪ᚋ⸨ ▱ఙ Ꮫ⾡㞧ㄅྡ Dynamics and Design Conference 2012 Ⓨ⾜ᖺ᭶᪥ ᖹᡂ 24 ᖺ 9 ᭶ 19 ᪥
94
௜㘓
95
$ ㍈せ⣲ࡢ㉁㔞⾜ิ㸪๛ᛶ⾜ิ㸪ࢪࣕ࢖ࣟ⾜ิ
ᐦᗘ ȡ(kg/m3)㸪᩿㠃✚ Ae (m2)㸪᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ Ie (m4)㸪㛗ࡉ le(m)㸪ࡏࢇ᩿ಀᩘ IJe ࡢ
㍈せ⣲ࡢ㉁㔞⾜ิ㸪๛ᛶ⾜ิ㸪ࢪࣕ࢖ࣟ⾜ิࡣࡑࢀࡒࢀᘧ㸦A.1㸧㸪ᘧ㸦A.2㸧㸪ᘧ㸦A.3㸧࡛
ᐃ⩏ࡉࢀࡿ㸬
(1) ㉁㔞⾜ิ
M se
0
m3
0
m4 º
0
0
m2
ª m1
« 0
0
m3 m4
0 »»
m1 m2 0
«
0»
m6
0
m5
0
m4
« 0 m2
«
UAe le
m5 m4
0
0
m6 »
m2
0
0
«
»
2
0
0
0 m2 »
m1
0 m4
8401 + W e « m3
« 0
m3
0
0 »
m1 m2
0
m4
« 0 m
»
0
m
0
m
0
m
5
2
6
4
«
»
0
m5 »¼
0
m6 m 2
0
0
«¬ m4
㸦A.1㸧
0
0
0
0
m7
m8
m8 º
ª m7
« 0
0 m7 m8
0 »»
m7 m8 0
«
0
0
0 »
m9
m8
m10
« 0 m8
«
UI e
0
0
0
0
m
m9 m8
m10 »
»
« 8
2 m
0
0 m8
0
0 m8 »
m7
30l e 1 + W e « 7
« 0
0
0
0 »
m7
m8
m7
m8
« 0 m
0
0
0 »
m10
m8
m9
8
»
«
0
0
0
0
m8
m10
m9 »¼
«¬ m8
m1
312 588W e 280W 2
m2
44 77W
e
m3
108 252W e 140W 2
m4
26 63W e 35W e ˜ l e
35W 2 ˜ l e
m6
8 14W 7W ˜ l
6 14W 7W ˜ l
m7
36
m5
m8
m9
m10
2
2
e
2
e
e
2
e
e
2
e
3 15W e ˜ le
4 5W 10W ˜ l
1 5W 5W ˜ l
2
e
e
2
e
2
e
e
2
e
96
(2) ๛ᛶ⾜ิ
K se
EI e
1 + W e ˜ le 3
0
ª k1
« 0
k1
«
« 0 k2
« k
0
« 2
k
0
1
«
« 0 k1
« 0 k
2
«
0
¬« k 2
k1
12
k2
6l e
k3
4 W e ˜ le 2
2 W e ˜ le 2
k4
k2
0
k2
0
k3 0
k3
0
0 k2
k2
k4
0
0
0
k4
k1
0
0 k1
k2
0
k2
0
k1
0
k1
0
k2
0
k2
0
0
k2
k4
0
0
k2
k3
0
k2 º
0 »»
0»
k4 »
» 㸦A.2㸧
k2 »
0»
0 »
»
k 3 ¼»
(3) ࢪࣕ࢖ࣟ⾜ิ
Gse
g1
g2
g3
g4
UI e
15le 1 + W e 2
0
g2
ª 0 g1
« g
0
0
g2
« 1
0
g3
0
g
«
2
« 0 g
0
g3
2
«
0
0
g
g
1
2
«
« g1
0
0 g2
« g
0
0
g4
« 2
0
g4
«¬ 0 g 2
36
3 15W e ˜ le
4 5W e 10W e ˜ le 2
1 5W
2
e
5W e ˜ l e
2
97
0
g1
g2
0
g1 0
0
0
g2
0
g2
g4
0 g1 g 2
0
0
g1
0
0
g2
g3
0
g2
0 º
g 2 »»
g4 »
0 »
» 㸦 A.3 㸧
0 »
g2 »
g3 »
»
0 »¼
$ ᩿㠃୍ᵝᱱࡢ᭤ࡆ᣺ືࡢᅛ᭷᣺ືᩘ
᩿㠃୍ᵝᱱࡢ᭤ࡆ᣺ືࡢᅛ᭷᣺ືᩘࢆ㸪࢜࢖࣮࣭ࣛ࣋ࣝࢾ࣮࢖ᱱ⌮ㄽ㸪ࢸ࢕ࣔࢩ࢙ࣥࢥ
ᱱ⌮ㄽ㸪࠾ࡼࡧࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽࢆ㐺⏝ࡋࡓ 1 ḟඖᱱࡢ᭷㝈せ⣲ἲࡢ 3 ࡘࡢ᪉ἲ࡟ࡘ
࠸࡚ẚ㍑ࡍࡿ㸬࢜࢖࣮࣭ࣛ࣋ࣝࢾ࣮࢖ᱱ⌮ㄽ࡜ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽ࡜ࡢᕪ␗ࡣ㸪ࢸ࢕ࣔ
ࢩ࢙ࣥࢥᱱ⌮ㄽ࡛ࡣ㸪࢜࢖࣮࣭ࣛ࣋ࣝࢾ࣮࢖ᱱ⌮ㄽ࡛ࡣ⪃៖ࡉࢀ࡞࠸㸪ࡏࢇ᩿ኚᙧ࡜ᅇ㌿
័ᛶࡀ⪃៖ࡉࢀࡿⅬ࡛࠶ࡿ㸬
(1) ィ⟬ᑐ㇟
ࡇࡇ࡛ࡣ㸪➨ 3 ❶ࡢᅗ 3.10 ࡟♧ࡉࢀࡿ㸪୧➃⮬⏤ࡢᨭᣢ᮲௳ࡢ୍ᵝ࡞୰ᐇ෇ᙧ᩿㠃㍈ࢆ
ྲྀࡾୖࡆ࡚㸪ୖ㏙ࡢ 3 ࡘࡢ᪉ἲ࡛ồࡵࡓ 1 ḟ࠿ࡽ 4 ḟࡲ࡛ࡢ᭤ࡆ᣺ື࣮ࣔࢻࡢᅛ᭷᣺ືᩘ
ࢆẚ㍑ࡍࡿ㸬௨ୗ࡟୰ᐇ෇ᙧ᩿㠃㍈ࡢᮦᩱ≉ᛶ್࡜ᑍἲࢆ♧ࡍ㸬
ᮦᩱ≉ᛶ㸸ࣖࣥࢢ⋡ E=2.09×1011 N/m2㸪ᶓᙎᛶಀᩘ G= 8.0335×1010 N/m2㸪
࣏࢔ࢯࣥẚ Ȟ=0.3008㸪ᐦᗘ ȡ=7846 kg/m3
ᑍ ἲ 㸸඲㛗 L=350mm㸪┤ᚄ d=20mm
ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽࢆ㐺⏝ࡋࡓ 1 ḟඖᱱࡢ᭷㝈せ⣲ἲ࡛ࡣ㸪ᅗ 3.11 ࡟♧ࡉࢀࡿ⠇Ⅼᩘ
35 ࡢࣔࢹࣝࢆ⏝࠸ࡿ㸬ྛ⠇Ⅼࡢ㍈᪉ྥࡢᗙᶆࡣ
0㸪14.35㸪28.7㸪40.1875㸪51.675㸪57.3㸪63.65㸪70㸪76.35㸪82.7㸪88.325㸪98.7625㸪109.2㸪
122.5㸪135.8㸪146.2375㸪156.675㸪162.3㸪168.65㸪175㸪181.35㸪187.7㸪193.325㸪203.562㸪
213.8㸪226.9㸪240㸪253.05㸪266.1㸪279.15㸪292.2㸪306.65㸪321.1㸪335.55㸪350
࡛࠶ࡿ㸦༢఩㸸mm㸧㸬
(2) ᅛ᭷᣺ືᩘࡢᑟฟ
࢜࢖࣮࣭ࣛ࣋ࣝࢾ࣮࢖ᱱ⌮ㄽ࡟࠾࠸࡚㸪ቃ⏺᮲௳ࡀ୧➃⮬⏤ࡢ᩿㠃୍ᵝᱱࡢ᭤ࡆ᣺ືࡢ
᣺ືᩘ᪉⛬ᘧࡣᘧ㸦A.4㸧࡛⾲ࡉࢀ㸪ᅛ᭷᣺ືᩘ fn ࡣᘧ㸦A.5㸧࡛୚࠼ࡽࢀࡿ㸦Seon, M., et al.,
1999㸧㸬
cos ka ˜ cosh ka 1 0 㸦A.4㸧
ka㸸↓ḟඖἼᩘ
98
EI
2
k (Hz) 㸦A.5㸧
4 a
UAL
1
2S
fn
I㸸᩿㠃 2 ḟ࣮࣓ࣔࣥࢺ(m4)㸪 I
Sd 4 64
A㸸᩿㠃✚(m2)㸪 A Sd 2 4
ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽ࡟࠾࠸࡚㸪ቃ⏺᮲௳ࡀ୧➃⮬⏤ࡢ᩿㠃୍ᵝᱱࡢ᭤ࡆ᣺ືࡢ᣺ືᩘ
᪉⛬ᘧࡣᘧ㸦A.6㸧࡛⾲ࡉࢀ㸪ᘧ㸦A.6㸧୰ࡢ ka ࡜ kb ࡢ㛵ಀࡣᘧ㸦A.7㸧࡛୚࠼ࡽࢀࡿ㸬ᅛ᭷
᣺ືᩘ fn ࡣᘧ㸦A.8㸧࡛୚࠼ࡽࢀࡿ㸦Seon, M., et al., 1999㸧㸬
k
2
a
k b k a kb [ 2 k a k b k a kb k a kb [ 2 k a kb k a k b
˜ sin ka ˜ sinh kb cos ka ˜ cosh kb 1 0
2
2
2
2 k a k b k b [ 2 k a k a [ 2 kb
2
2
2
2
2
kb㸸ィ⟬ୖࡢ↓ḟඖἼᩘ 㸦A.6㸧
[
2
NG
k [ k k k 1 [ 2
kb ka
2
2
a
2
2
㸦A.7㸧
(Hz) 㸦A.8㸧
b
2
2
b
A I ˜L
s㸸⦪ᶓẚ㸪 s
1
2SL
s2 a
2
fn
21 Q 7 6Q 61 Q E
ȟ2㸸ಀᩘ㸪 [ 2
2
E ka kb
U 1 [ 2
2
ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽࢆ㐺⏝ࡋࡓ 1 ḟඖᱱࡢ᭷㝈せ⣲ἲ࡛ࡣ㸪௜㘓 A-1 ࡟♧ࡉࢀࡓ᪉ἲ
࡛㉁㔞⾜ิ[M]࡜๛ᛶ⾜ิ[K]ࡀᐃ⩏ࡉࢀ㸪ᘧ㸦A.9㸧ࡢᐇᅛ᭷್ၥ㢟ࢆゎࡃࡇ࡜࡟ࡼࡗ࡚ᅛ
᭷᣺ືᩘ fn ࢆồࡵࡿࡇ࡜ࡀ࡛ࡁࡿ㸬
>K @ O>M @^)` ^O` Ȝ㸸ᅛ᭷್㸪 O
㸦A.9㸧
2Sf n 2
{ĭ}㸸ᅛ᭷࣋ࢡࢺࣝ
(3) ィ⟬⤖ᯝ
࢜࢖࣮࣭ࣛ࣋ࣝࢾ࣮࢖ᱱ⌮ㄽ࡟࠾࠸࡚㸪ᘧ㸦A.4㸧ࡢ᣺ືᩘ᪉⛬ᘧ࠿ࡽᚓࡽࢀࡿ 1 ḟ࠿ࡽ
4 ḟࡲ࡛ࡢ ka ࡢ್ࡣ㸪
99
ka 1=4.7300407
ka 2=7.8532046
ka 3=10.995608
ka 4=14.137165
࡛࠶ࡿ㸦ῧ࠼ᏐࡢᩘᏐࡣ࣮ࣔࢻḟᩘࢆ♧ࡍ㸧㸬
ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽ࡟࠾࠸࡚㸪ᘧ㸦A.6㸧ࡢ᣺ືᩘ᪉⛬ᘧ࠿ࡽᚓࡽࢀࡿ 1 ḟ࠿ࡽ 4 ḟࡲ
࡛ࡢ ka ࡜ kb ࡢ್ࡣ㸪
ka 1=4.7306384㸪kb 1=4.6886925
ka 2=7.8526405㸪kb 2=7.6650678
ka 3=10.994838㸪kb 3=10.496361
ka 4=14.135824㸪kb 4=13.118302
࡛࠶ࡿ㸦ῧ࠼ᏐࡢᩘᏐࡣ࣮ࣔࢻḟᩘࢆ♧ࡍ㸧㸬
ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽ㸪࢜࢖࣮࣭ࣛ࣋ࣝࢾ࣮࢖ᱱ⌮ㄽ㸪ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽࢆ㐺⏝ࡋ
ࡓ 1 ḟඖᱱࡢ᭷㝈せ⣲ἲࡢ 3 ࡘࡢ᪉ἲ࡛ồࡵࡓ㸪୰ᐇ෇ᙧ᩿㠃㍈ࡢ 1 ḟ࠿ࡽ 4 ḟࡲ࡛ࡢ᭤
ࡆ᣺ືࡢᅛ᭷᣺ືᩘࢆ௨ୗ࡟♧ࡍ㸬࢜࢖࣮࣭ࣛ࣋ࣝࢾ࣮࢖ᱱ⌮ㄽ࡜ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽ
ࢆ㐺⏝ࡋࡓ 1 ḟඖᱱࡢ᭷㝈せ⣲ἲ࡟ࡼࡿᅛ᭷᣺ືᩘࡢィ⟬⤖ᯝ࡟ࡣ㸪ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮
ㄽ࡟ࡼࡿィ⟬⤖ᯝ࡜ࡢᕪ␗ࢆⓒศ⋡࡛ేグࡋࡓ㸬
Natural frequencies of bending vibration modes of uniform beam
Unit: Hz
Timoschenko Euler-Bernoulli beam
1D FEM
Mode beam theory
theory
(Error)
(Error)
1
743.68
750.12
(0.87%)
743.69
(0.00%)
2
2018.5
2067.7
(2.44%)
2018.8
(0.01%)
3
3872.4
4053.6
(4.68%)
3873.7
(0.03%)
4
6230.5
6700.8
(7.55%)
6236.7
(0.10%)
ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽ࡟ᑐࡍࡿ࢜࢖࣮࣭ࣛ࣋ࣝࢾ࣮࢖ᱱ⌮ㄽࡢᅛ᭷᣺ືᩘࡢᕪ␗ࡣ㸪ࣔ
࣮ࢻḟᩘࡀ㧗ࡃ࡞ࡿ࡯࡝኱ࡁࡃ࡞ࡾ㸪4 ḟ࣮ࣔࢻ࡛ࡣ 7.55%ࡢᕪ␗ࡀ⏕ࡌ࡚࠸ࡿ㸬୍᪉㸪ࢸ
࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽࢆ㐺⏝ࡋࡓ 1 ḟඖᱱࡢ᭷㝈せ⣲ἲ࡛ࡣ㸪ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽ࡟ᑐࡋ
࡚ 4 ḟ࣮ࣔࢻࡲ࡛࡛ࡓ࠿ࡔ࠿ 0.1%ࡢᅛ᭷᣺ືᩘࡢᕪ␗ࡀ⏕ࡌࡿ࡟ࡍࡂ࡞࠸㸬
100
ࡲࡓ㸪ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽࢆ㐺⏝ࡋࡓ 1 ḟඖᱱࡢ᭷㝈せ⣲ࣔࢹࣝࡢ⠇Ⅼᩘ 35 ࡣ㸪╔┠
ࡋࡓ᭱኱ࡢ࣮ࣔࢻḟᩘ 4 ࡢ 8.75 ಸ࡛࠶ࡿ㸬
ࡼࡗ࡚㸪୍ᵝ᩿㠃ࡢ㍈ࡢ᭤ࡆ᣺ືࡢᅛ᭷᣺ືᩘࢆᢅ࠺ሙྜ㸪╔┠ࡍࡿ᭱኱࣮ࣔࢻḟᩘࡢ
10 ಸ⛬ᗘࡢᩘࡢ⠇Ⅼࢆ࡛ࡁࡿࡔࡅᆒ➼࡟タࡅࡓ㸪ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽࢆ㐺⏝ࡋࡓ 1 ḟඖ
ᱱࡢ᭷㝈せ⣲ࣔࢹࣝࢆ⏝࠸ࡿࡇ࡜࡟ࡼࡗ࡚㸪ࢸ࢕ࣔࢩ࢙ࣥࢥᱱ⌮ㄽ࡟ࡼࡿࡶࡢ࡜࡯ࡰྠ୍
ࡢ⤖ᯝࢆᚓࡿࡇ࡜ࡀ࡛ࡁࡿ࡜ゝ࠼ࡿ㸬
ཧ⪃ᩥ⊩
Seon, M., Han, H. B., and Timothy, W., Dynamics of transversely vibrating beams using for
engineering theories, Journal of Sound and Vibration, Vol. 255, No.5 (1999), pp. 935-988.
101
$ ㍈ࡢ〇సᅗ
ᅗ
102
$ ࢱ࣮ࣅࣥ㸫Ⓨ㟁ᶵ࣮ࣟࢱࡢ᭤ࡆ᣺ືゎᯒࣔࢹࣝࡢㅖඖ
Coordinate of nodes for bending vibration analysis model of turbine – generator rotor
Node
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
z (m)
0.000
0.390
0.481
0.692
0.780
1.300
1.623
1.755
1.991
2.210
2.730
3.185
3.614
3.900
4.160
4.420
4.680
4.940
5.200
5.460
5.705
5.990
6.058
6.292
6.526
6.760
7.280
7.447
7.755
7.839
8.151
8.450
8.767
8.840
9.133
9.230
9.620
10.075
10.231
10.524
Node
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
z (m)
10.628
10.847
11.001
11.133
11.375
11.700
12.025
12.155
12.448
12.608
12.740
12.935
13.130
13.262
13.507
13.884
14.261
14.638
14.883
15.015
15.210
15.405
15.540
15.860
16.328
16.705
17.076
17.311
17.446
17.914
18.044
18.564
19.084
19.348
19.604
19.905
20.205
20.506
20.807
21.107
103
Node
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
-
z (m)
21.408
21.708
22.009
22.310
22.610
22.911
23.212
23.512
23.813
24.113
24.414
24.670
24.947
25.480
26.000
26.130
26.650
26.785
27.060
27.469
27.879
28.288
28.379
28.711
29.133
29.188
29.374
29.614
-
Dimensions of shaft for bending vibration analysis model of turbine – generator rotor
( Density 7850 kg/m3, Young's modulus 7850 Pa㸪Shear modulus 7.9192™1010 Pa )
Node
Outer
Diameter
(m)
Node
Outer
Diameter
(m)
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
6 to 7
7 to 8
8 to 9
9 to 10
10 to 11
11 to12
12 to 13
13 to 14
14 to 15
15 to 16
16 to 17
17 to 18
18 to 19
19 to 20
20 to 21
21 to 22
22 to 23
23 to 24
24 to 25
25 to 26
26 to 27
27 to 28
28 to 29
29 to 30
30 to 31
31 to 32
32 to 33
33 to 34
34 to 35
35 to 36
36 to 37
37 to 38
38 to 39
39 to 40
40 to 41
0.312
0.689
0.312
0.312
0.468
0.468
0.468
0.780
0.780
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.092
1.092
1.092
1.092
1.092
1.092
1.092
1.300
1.300
1.300
1.300
1.300
1.300
1.508
1.508
1.508
1.508
1.040
1.040
41 to 42
42 to 43
43 to 44
44 to 45
45 to 46
46 to 47
47 to 48
48 to 49
49 to 50
50 to 51
51 to 52
52 to 53
53 to 54
54 to 55
55 to 56
56 to 57
57 to 58
58 to 59
59 to 60
60 to 61
61 to 62
62 to 63
63 to 64
64 to 65
65 to 66
66 to 67
67 to 68
68 to 69
69 to 70
70 to 71
71 to 72
72 to 73
73 to 74
74 to 75
75 to 76
76 to 77
77 to 78
78 to 79
79 to 80
80 to 81
0.780
0.780
0.468
0.468
0.468
0.468
0.858
0.468
0.468
0.468
0.910
0.910
0.468
0.468
0.468
0.468
0.468
0.468
0.468
0.910
0.910
0.481
0.481
0.481
0.481
0.481
0.481
0.481
0.910
1.300
0.910
0.910
0.910
0.910
1.170
1.170
1.170
1.170
1.170
1.170
104
Node
81 to 82
82 to 83
83 to 84
84 to 85
85 to 86
86 to 87
87 to 88
88 to 89
89 to 90
90 to 91
91 to 92
92 to 93
93 to 94
94 to 95
95 to 96
96 to 97
97 to 98
98 to 99
99 to 100
100 to 101
101 to 102
102 to 103
103 to 104
104 to 105
105 to 106
106 to 107
107 to 108
-
Outer
Diameter
(m)
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
0.910
0.910
0.910
0.910
1.300
0.910
0.481
0.481
0.481
0.481
0.481
0.325
0.325
0.325
0.195
0.195
0.195
-
Specification of disc elements for bending vibration analysis model of turbine – generator rotor
Node
Density
(kg/m3)
Thickness
(m)
Outer
diameter
(m)
Outer
diameter
(m)
Node
Density
(kg/m3)
Thickness
(m)
Outer
diameter
(m)
Outer
diameter
(m)
13
14
15
16
17
18
19
20
23
24
25
26
30
31
32
34
36
37
38
13
14
15
16
17
18
19
20
23
24
25
26
30
31
32
34
36
37
38
76
77
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7222
7222
0.130
0.130
0.130
0.130
0.130
0.130
0.130
0.130
0.135
0.135
0.135
0.135
0.169
0.169
0.169
0.169
0.195
0.195
0.312
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.009
0.009
0.009
0.009
0.012
0.012
0.012
0.012
0.016
0.017
0.019
0.301
0.301
1.404
1.404
1.404
1.404
1.404
1.404
1.404
1.404
1.508
1.508
1.508
1.508
1.820
1.820
1.820
1.820
1.820
1.820
1.820
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.820
1.872
1.950
2.080
2.444
2.600
2.691
2.860
3.055
3.315
3.640
1.580
1.580
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.092
1.092
1.092
1.092
1.300
1.300
1.300
1.300
1.508
1.508
1.508
1.404
1.404
1.404
1.404
1.404
1.404
1.404
1.404
1.508
1.508
1.508
1.508
1.820
1.820
1.820
1.820
1.820
1.820
1.820
1.170
1.170
78
79
80
81
82
83
84
85
86
87
88
89
90
-
7222
7222
7222
7222
7222
7222
7222
7222
7222
7222
7222
7222
7222
-
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
-
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
-
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
-
105
$ ࢱ࣮ࣅࣥ㸫Ⓨ㟁ᶵ࣮ࣟࢱࡢࡡࡌࡾ᣺ືゎᯒࣔࢹࣝࡢㅖඖ
Coordinate of nodes for torsional vibration analysis model of turbine – generator rotor
Node
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
z (m)
0.000
0.390
0.481
0.739
0.780
1.300
1.691
1.755
2.114
2.210
2.730
3.185
3.614
3.900
4.160
4.420
4.680
4.940
5.200
5.460
5.884
5.990
6.058
6.292
6.526
6.760
7.280
7.629
7.755
7.839
8.151
8.450
8.840
8.986
9.133
9.230
9.620
10.075
10.231
10.366
Node
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
z (m)
10.628
10.725
11.001
11.065
11.375
11.700
12.025
12.155
12.448
12.674
12.740
12.935
13.130
13.196
13.507
13.884
14.261
14.638
14.949
15.015
15.210
15.405
15.472
15.860
16.328
16.705
17.076
17.379
17.446
17.914
18.044
18.564
19.084
19.493
19.604
19.905
20.205
20.506
20.807
21.107
106
Node
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
-
z (m)
21.408
21.708
22.009
22.310
22.610
22.911
23.212
23.512
23.813
24.113
24.414
24.525
24.947
25.480
26.000
26.130
26.650
26.717
27.060
27.469
27.879
28.288
28.330
28.711
29.133
29.159
29.374
29.614
-
Dimensions of Shaft for torsional vibration analysis model of turbine – generator rotor
( Density 7850 kg/m3, Shear modulus 7.9192™1010 Pa )
Node
Outer
Diameter
(m)
Node
Outer
Diameter
(m)
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
6 to 7
7 to 8
8 to 9
9 to 10
10 to 11
11 to 12
12 to 13
13 to 14
14 to 15
15 to 16
16 to 17
17 to 18
18 to 19
19 to 20
20 to 21
21 to 22
22 to 23
23 to 24
24 to 25
25 to 26
26 to 27
27 to 28
28 to 29
29 to 30
30 to 31
31 to 32
32 to 33
33 to 34
34 to 35
35 to 36
36 to 37
37 to 38
38 to 39
39 to 40
40 to 41
0.312
0.689
0.312
0.312
0.468
0.468
0.468
0.780
0.780
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.092
1.092
1.092
1.092
1.092
1.092
1.092
1.300
1.300
1.300
1.300
1.300
1.300
1.508
1.508
1.508
1.508
1.040
1.040
41 to 42
42 to 43
43 to 44
44 to 45
45 to 46
46 to 47
47 to 48
48 to 49
49 to 50
50 to 51
51 to 52
52 to 53
53 to 54
54 to 55
55 to 56
56 to 57
57 to 58
58 to 59
59 to 60
60 to 61
61 to 62
62 to 63
63 to 64
64 to 65
65 to 66
66 to 67
67 to 68
68 to 69
69 to 70
70 to 71
71 to 72
72 to 73
73 to 74
74 to 75
75 to 76
76 to 77
77 to 78
78 to 79
79 to 80
80 to 81
0.780
0.780
0.468
0.468
0.468
0.468
0.858
0.468
0.468
0.468
0.910
0.910
0.468
0.468
0.468
0.468
0.468
0.468
0.468
0.910
0.910
0.481
0.481
0.481
0.481
0.481
0.481
0.481
0.910
1.300
0.910
0.910
0.910
0.910
1.170
1.170
1.170
1.170
1.170
1.170
107
Node
81 to 82
82 to 83
83 to 84
84 to 85
85 to 86
86 to 87
87 to 88
88 to 89
89 to 90
90 to 91
91 to 92
92 to 93
93 to 94
94 to 95
95 to 96
96 to 97
97 to 98
98 to 99
99 to 100
100 to 101
101 to 102
102 to 103
103 to 104
104 to 105
105 to 106
106 to 107
107 to 108
-
Outer
Diameter
(m)
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
1.170
0.910
0.910
0.910
0.910
1.300
0.910
0.481
0.481
0.481
0.481
0.481
0.325
0.325
0.325
0.195
0.195
0.195
-
Specification of disc elements for torsional vibration analysis model of turbine – generator rotor
Node
Density
(kg/m3)
Thickness
(m)
Outer
diameter
(m)
Outer
diameter
(m)
Node
Density
(kg/m3)
Thickness
(m)
Outer
diameter
(m)
Outer
diameter
(m)
13
14
15
16
17
18
19
20
23
24
25
26
30
31
32
33
36
37
38
13
14
15
16
17
18
19
20
23
24
25
26
30
31
32
33
36
37
38
76
77
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7850
7222
7222
0.130
0.130
0.130
0.130
0.130
0.130
0.130
0.130
0.135
0.135
0.135
0.135
0.169
0.169
0.169
0.169
0.195
0.195
0.312
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.009
0.009
0.009
0.009
0.012
0.012
0.012
0.012
0.016
0.017
0.019
0.301
0.301
1.404
1.404
1.404
1.404
1.404
1.404
1.404
1.404
1.508
1.508
1.508
1.508
1.820
1.820
1.820
1.820
1.820
1.820
1.820
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.612
1.820
1.872
1.950
2.080
2.444
2.600
2.691
2.860
3.055
3.315
3.640
1.580
1.580
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.014
1.092
1.092
1.092
1.092
1.300
1.300
1.300
1.300
1.508
1.508
1.508
1.404
1.404
1.404
1.404
1.404
1.404
1.404
1.404
1.508
1.508
1.508
1.508
1.820
1.820
1.820
1.820
1.820
1.820
1.820
1.170
1.170
78
79
80
81
82
83
84
85
86
87
88
89
90
-
7222
7222
7222
7222
7222
7222
7222
7222
7222
7222
7222
7222
7222
-
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
0.301
-
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
1.580
-
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
1.17
-
108