東北地域災害科学研究

117

第59巻（2023）

័ᛶ㉁㔞ຠᯝࢆ᭷ࡍࡿᾮὶࢲࣥࣃ࣮ࢆ⏝࠸ࡓ㸱ᒙ㦵⤌ࡢ᣺ືᛂ⟅≉ᛶ㸦ࡑࡢ 㸰㸧 ᐇ㦂 ࠾ࡼ ࡧ᫬ ้Ṕᛂ ⟅ゎ ᯒ⤖ ᯝᑠ ᔱ೺ኴ 㑻 ᮾ ໭ ᕤ ᴗ ኱ Ꮫ ኱ Ꮫ 㝔 ᑠᮾ໭ᕤ ᴗ኱ Ꮫ 㸪 ⯪ ᮌᑦ ᕫ௝ྎ㧗 ᑓ 㣤⸨ ᑗஅ 㸪⸨ ⏣ᬛᕫ 㸪Ỉ 㛫ᕦ ኱1㸬
㸬 ࡣࡌࡵ ࡟ ᆅ㟈᫬࡟࠾ࡅࡿᘓ≀ࡢᛂ⟅ࢆᢚไࡍࡿࡇ࡜ࢆ┠ⓗ࡜ࡋ࡚័ᛶ㉁㔞ຠᯝࢆ᭷ࡍࡿᾮὶࢲࣥࣃ࣮ 1 㸧 ࢆᥦ᱌ࡋ㸪ࡑࡢຠᯝࢆ 1 ᒙࡢ⦰ᑠ㦵⤌ヨ㦂య࡟ࡼࡿ᣺
ືྎຍ᣺ヨ㦂࡜᫬้Ṕᛂ⟅ゎᯒ࡟ࡼࡾ᳨ドࡋ࡚ࡁࡓ ౛࠼ࡤ 2 㸧ࠋᮏሗ࡟࠾࠸࡚ࡣ㸪ᮏࢲࣥࣃ࣮ࡀ⤌ࡳ㎸ࡲࢀࡓ⦰ᑠ 3 ᒙ㦵⤌ヨ㦂య࡟ࡼࡿ᣺ືྎຍ᣺ヨ㦂ࡢ⤖ᯝࢆࢩ࣑࣮ࣗࣞࢩࣙࣥࡍࡿࡓࡵࡢ᫬้Ṕᛂ⟅ゎᯒࣉࣟࢢ࣒ࣛࢆసᡂࡋ㸪ᐇ㦂⤖ᯝ࡜ゎᯒ⤖ᯝࡢẚ㍑
࠿ࡽ㸪ゎᯒᡭἲࡢጇᙜᛶࢆ᳨ドࡍࡿࡇ࡜ࢆ୺࡞┠ⓗ࡜ࡍࡿࠋ2㸬㸬 ゎᯒࢆ ⾜ࡗ ࡓᑐ ㇟ᐇ 㦂ࡢ ᴫせ 
ࡑࡢ 1 ࡛♧ࡋࡓᐇ㦂ࢆᇶ࡟ゎᯒࢆ⾜ࡗࡓࠋᾮὶࢲ
ࣥࣃ࣮ࢆ 1 ᒙࡶࡋࡃࡣ 1 ᒙ࡜ 2 ᒙ࡟⤌ࡳ㎸ࢇ࡛ṇᘻἼධຊ࡛ຍ᣺ヨ㦂ࢆ⾜ࡗࡓࠋຍ᣺ࣞ࣋ࣝࡣ 25gal࡜ࡋ㸪1.0Hz ࠿ࡽ 0.5Hz ࡎࡘኚ໬ࡉࡏ㸪10Hz ࡲ࡛࡜ࡋࡓࠋ
3㸬㸬 ᫬้Ṕ ᛂ⟅ ゎᯒ ᅗ㸯ゎᯒࣔࢹࣝࢲࣥࣃ࣮ࡀ 1 ᒙ࡟㓄⨨ࡉࢀࡓሙྜࡢゎᯒࣔࢹࣝࢆᅗ 1 ࡟♧ࡍࠋゎᯒࣔࢹࣝࢆᇶ࡟ࡋࡓ㞳ᩓ᫬㛫⣔ࡢ᣺ື᪉⛬ᘧࡣ㸪̅�����
�[�̈ − ��� + �� �� ]{�̈ }� + [� ]{�� }� + [K]{�}� = −[�]{1}���

���(1)

� ]㸪[�̅ ]㸪
࡜ᑟ࠿ࢀࡿࠋ��̈ 㸪�� � 㸪�� ࡣࡑࢀࡒࢀḟࡢࢫࢸࢵࣉࡢຍ㏿ᗘࠊ㏿ᗘࠊኚ఩ࢆ⾲ࡋ࡚࠾ࡾ㸪[�� ]ࡣࡑࢀࡒࢀ㉁㔞࣐ࢺࣜࢡࢫ㸪ῶ⾶࣐ࢺࣜࢡࢫ㸪๛ᛶ࣐ࢺࣜࢡࢫࢆ⾲ࡋ࡚࠸ࡿࠋ[�����[K� ]ࡣ㉁㔞ຠᯝࡀ௜ຍࡉࢀࡓ㉁㔞࣐ࢺࣜࢵࢫࢆ⾲ࡋ㸪

Vibration Response Characteristics of Multi Story Frame Attached with Liquid Damper
By kentarou KOJIMA et al.

118

��
����
0[�]=��00��00� � ���0 �+� 0
��
00 00 0�
0 0(2)

࡜⾲ࡉࢀࡿࠋ(2)ᘧ࡟࠾࠸࡚㸪�� ࡣ㐃⤖⟶ෆࡢᾮయ㉁㔞ࢆ㸪�ࡣࣆࢫࢺࣥࡢ᭷ຠ᩿㠃✚�࡜㐃⤖⟶ࡢ᩿㠃✚�ࡢẚࢆ⾲ࡋ㸪� � �� ࡀ㐃⤖⟶ෆࡢᾮὶ࡟ࡼࡿ㉁㔞ຠᯝࢆ♧ࡋ࡚࠸ࡿࠋ(1)ᘧࡢྑ㎶➨ 1㡯ࡢ�� � ࡣᆅືຍ㏿ᗘࢆ㸪➨ 2 㡯ࡢ�� ࡜�� ࡣ㸪ࡑࢀࡒࢀࢩ࣮ࣝᮦ࡟ࡼࡿ᢬ᢠຊ࡜ᾮὶ࡟ࡼࡿ᢬ᢠ

ຊࢆ⾲ࡋ࡚࠾ࡾ㸪(3)ᘧ㸪(4)ᘧࡢࡼ࠺࡟⾲ࡉࢀࡿࠋࡲࡓ㸪ࢲࣥࣃ࣮ࡢᴫᛕᅗࢆᅗ 2 ࡟♧ࡍࠋ�� = �� �̇ + �� �

�� = (��� + ��� + ���) × �(3)

(4)

(3)ᘧ࡟࠾࠸࡚㸪�� 㸪�� ࡣࡑࢀࡒࢀ㸪ࢩ࣮ࣝᮦࡢῶ⾶ಀᩘ࡜๛ᛶࢆ♧ࡋ㸪�̇ 㸪�ࡣࡑࢀࡒࢀࣆࢫࢺ

ࣥࡢ㏿ᗘ࡜ኚ఩ࢆ⾲ࡋ࡚࠸ࡿࠋ(4)ᘧ࡟࠾ࡅࡿ ��� 㸪���㸪���ࡣ㐃⤖⟶ෆࡢᅽຊᦆኻ࡛㸪ὶ㊰ᙧ≧ࡀᛴ⃭࡟ኚ໬ࡍࡿᛴ⦰ᑠ㒊࡜ᛴᣑ኱㒊ࡢ⟬ฟᘧࡣࡑࢀࡒࢀ㸪��� = � ∙�(� ∙ �̇ )�� ∙ �̇ �= � ∙ �� ∙22

��� =�(�∙�̇ )�
�࡜⾲ࡉࢀࡿࠋࡲࡓ㸪㐃⤖⟶ෆᦶ᧿࡟ࡘ࠸࡚ࡣ㸪࡜⾲ࡉࢀࡿࠋ��� == �� ∙�∙�̇ ���∙� �� ∙�
2� �(5)(6)(7)ᅗ㸰 ࢲࣥࣃ࣮ᴫᛕᅗ(5)ᘧ࡟࠾࠸࡚�㸦=0.5㸧ࡣὶ㊰ᙧ≧࡟ࡼࡾᐃࡲࡿಀᩘࢆ⾲ࡋ࡚࠸ࡿࠋࡲࡓ㸪(5)ᘧ࠿ࡽ(7)ᘧ࡟࠾
࠸࡚ᾮయࡢᐦᗘࢆ �㸦=998.2kg/m3㸧࡜ࡋ࡚࠾ࡾ㸪 �ࡣ㐃⤖⟶㛗ࡉ㸪�ࡣ㐃⤖⟶ෆᚄ࡛࠶ࡾ㸪್ࢆ⾲㸯࡟♧ࡍࠋ�� ࡣ㐃⤖⟶ෆࢆὶࢀࡿὶయࡢᖹᆒ㏿ᗘࢆ♧ࡋ࡚࠾ࡾ㸪�� (= � ∙ �̇ � )

(8)

119

࡜⾲ࡉࢀࡿࠋ(7)ᘧࡢ㸪�ࡣ⟶ᦶ᧿ಀᩘࢆ⾲ࡋ࡚࠾ࡾ㸪ࢲࣥࣃ࣮༢య࡟ࡼࡿຍ᣺ヨ㦂ࡢ⤖ᯝ࠿ࡽᐇ㦂ᘧ࡜ࡋ࡚ᑟ࠿ࢀࡿ㸦⾲㸯㸧ࠋ࡞࠾ࠊࣞ࢖ࣀࣝࢬᩘ��ࡣࠊ
�� = ��� 㺃��࡜⾲ࡍࡇ࡜ࡀฟ᮶ࡿࠋࡇࡇ࡛�ࡣື⢓ᛶಀᩘࢆ♧ࡋ࡚࠾ࡾ㸪�ࡣᐊ
ࡓࠋ

( 9)

࡜ྠࡌ 20Υࡢ್ࢆ᥇⏝ࡋ4㸬
㸬 ゎᯒࣉ ࣟࢢ ࣒ࣛ ゎᯒ࡟ࡣᩘ್ゎᯒࢯࣇࢺࡢ MATLAB ࢆ౑⏝ࡋࡓࠋ(1)ᘧ࡟ࡘ࠸࡚ᩘ್✚ศἲ࡜ࡋ࡚ RungeKutta ἲࢆ㐺⏝ࡋ㸪ᛂ⟅ゎᯒࣉࣟࢢ࣒ࣛࢆసᡂࡋࡓࠋゎᯒࡣࢲࣥࣃ࣮ࡀ௜࠿࡞࠸ࣇ࣮࣒ࣞࡢࡳࡢ≧ែ㸦Case1㸧࡜ࢩ࣮ࣝᮦࡢࡳࡢࢲࣥࣃ࣮ࢆ⤌ࡳ㎸ࢇࡔࡶࡢ㸦Case2㸧㸪ࢲࣥࣃ࣮ࢆ㸯ᒙ࡟⤌ࡳ㎸ࢇࡔࡶࡢ㸦Case4 ࠿ࡽ Case6㸧࡟ࡘ࠸࡚⾜ࡗࡓࠋゎᯒࣉࣟࢢ࣒ࣛࡢࣇ࣮ࣟࢳ࣮ࣕࢺࢆᅗ㸱࡟㸪ゎᯒ࡟౑⏝ࡋࡓࣃ࣓࣮ࣛࢱࢆ⾲㸯࡟♧ࡍࠋࣇ࣮࣒ࣞࡢࣃ࣓࣮ࣛࢱࡣ Case1 ࡢ⮬⏤᣺ືヨ㦂ࢆ⾜ࡗ࡚ᚓࡽࢀࡓ⤖ᯝ࡟ᇶ࡙ࡁ⟬ฟࡋࡓ್ࢆ᥇⏝ࡋ࡚࠸ࡿࠋࢲࣥࣃ࣮ࡢࣃ࣓࣮ࣛࢱ࡟ࡘ࠸࡚ࡣ㸪ࢲࣥࣃ࣮༢యࡢຍ᣺ヨ㦂࠿ࡽᚓࡽࢀࡓᐇ㦂⤖ᯝ 3㸧 ࢆཧ⪃࡟ࡋࡓࠋ⾲㸯 ゎᯒࣃ࣓࣮ࣛࢱ
ࣇ࣮࣒ࣞࡢࣃ࣓࣮ࣛࢱ
㉁㔞
㹙��㹛
๛ᛶ
㹙 �/�㹛

ࣇ࣮ࣟࢳ࣮ࣕࢺ

44497.2 −22248.6
0�−22248.6 44497.2 −22248.6�
0−22248.6 22248.6

�� [�] + �� [�]㸦࣮࣮ࣞࣜᆺῶ⾶㸧�� , �� ࡣ 1.2%ࢲࣥࣃ࣮ࡢࣃ࣓࣮ࣛࢱ
0.0220ࣆࢫࢺࣥ᭷ຠ༙ᚄ㹙�㹛�
0.00152ࣆࢫࢺࣥ᭷ຠ᩿㠃✚㹙� 㹛0.01
㐃⤖⟶ෆᚄ㹙�㹛㐃⤖⟶㛗ࡉ㹙�㹛
0.5ࠊ1.0ࠊ2.0
19.36
᩿㠃✚ẚ
ࢲࣥࣃ࣮᢬ᢠຊࡢࣃ࣓࣮ࣛࢱ
ࡤࡡᐃᩘ
52432.2
㹙 �/�㹛
ࢩ࣮ࣝᮦ
ῶ⾶ᐃᩘ
66.9
㹙 � � ���/�㹛� = 0.5�� = 10611����.���⟶ෆᦶ᧿� = 1.0�� = 1219.3����.��ᐇ㦂ᘧ(� = 10��)� = 2.0�� = 926.94����.���
ῶ⾶ᐃᩘ
㹙㸣㹛ᅗ㸱

34.903800
� 0
28.7038
0 �
00
23.5164

1205㸬㸬 ᐇ㦂⤖ᯝ࡜ゎᯒ⤖ᯝࡢẚ㍑
ᐇ㦂⤖ᯝ࡜ゎᯒ⤖ᯝࢆẚ㍑ࡋࡓ௦⾲ⓗ࡞ඹ᣺᭤⥺ࡢ౛ࢆᅗ㸲࠿ࡽᅗ㸴࡟♧ࡍࠋᅗ㸲ࡣࢲࣥࣃ࣮ࡢ↓࠸ヨ㦂యࡢゎᯒ⤖ᯝ࡛࠶ࡾ㸪㸯ḟ㸪㸰ḟ㸪㸱ḟ࡜ࡶᐇ㦂⤖ᯝ࡜࡯ࡰྠࡌ
࿘Ἴᩘ࡛ᛂ⟅ಸ⋡ࡢࣆ࣮ࢡࡀᚓࡽࢀ࡚࠸ࡿࠋᛂ⟅ಸ⋡ࡢᑐᛂࡣ୙༑ศ࡛࠶ࡿࡀ㸪ᴫࡡ᣺ື≉ᛶࢆ⾲⌧ࡋ࡚࠸ࡿࠋ
ᅗ㸳ࡣ㸯ᒙ࡟ࢲࣥࣃ࣮ࢆྲྀࡾ௜ࡅᾮయࢆධࢀ࡚࡞࠸ヨ㦂యࡢゎᯒ⤖ᯝ࡛࠶ࡾ㸪ࢩ࣮ࣝᮦ࡟ࡼ
ࡿ㸯ḟࣆ࣮ࢡࢩࣇࢺࢆ⢭ᗘࡼࡃ෌⌧ࡋ࡚࠸ࡿࠋ㸰ḟ㸪㸱ḟ࡟㛵ࡋ࡚ࡣᐇ㦂⤖ᯝࡼࡾᰂࡽ࠿ࡵࡢ
࿘Ἴᩘ࡛ࣆ࣮ࢡࢆ♧ࡋ࡚࠸ࡿࡀ㸪ࢲࣥࣃ࣮᢬ᢠຊࡢࣃ࣓࣮ࣛࢱࡣᴫࡡⰋ࠸⢭ᗘ࡜⪃࠼ࡽࢀࡿࠋ
ᅗ㸴ࡣ㸯ᒙࡢࢲࣥࣃ࣮࡟ᾮయࢆᑒධࡋࡓヨ㦂యࡢゎᯒ⤖ᯝ࡛࠶ࡾ㸪㸯ḟࣆ࣮ࢡࡣᑐᛂࡀࡼ࠸
ࡶࡢࡢ㸰ḟ㸪㸱ḟࡣࣆ࣮ࢡࡀ෌⌧࡛ࡁࡎ㸪ඹ᣺࿘Ἴᩘ௜㏆࡛ᛂ⟅ಸ⋡࡟ẁᕪࡀࡘࡃ⤖ᯝ࡜࡞ࡗࡓࠋ
6㸬
㸬 ࡲ࡜ࡵ ័ᛶ㉁㔞ຠᯝࢆ᭷ࡍࡿᾮὶࢲࣥࣃ࣮ࡀ⤌ࡳ㎸ࡲࢀࡓ㸱ᒙ㦵⤌ヨ㦂యࢆᑐ㇟࡟㸪ࡑࡢືⓗ≉ᛶࢆゎᯒⓗ࡟ᢕᥱࡍࡿࡓࡵ㸪᫬้Ṕᛂ⟅ゎᯒࣉࣟࢢ࣒ࣛࢆᵓ⠏ࡋࡓࠋᮏሗ࡟࠾࠸࡚ࡣ㸪ᵓ⠏ࡋࡓゎᯒᡭἲࡢጇᙜᛶࢆ᳨ドࡍࡿࡓࡵ࡟㸪ゎᯒ⤖ᯝࢆ᣺ືྎຍ᣺ヨ㦂࡟ࡼࡾᚓࡽࢀࡓ⤖ᯝ࡜ẚ㍑ࡋ
ࡓࠋࡑࡢ⤖ᯝ㸪ࢲࣥࣃ࣮ࢆ㸯ᒙ࡟㓄⨨ࡋࡓሙྜ࡟ࡘ࠸࡚ࡣ㸪ᵓ⠏ࡋࡓゎᯒࣉࣟࢢ࣒ࣛ࡟ࡼࡿ⤖ᯝࡣᐇ㦂⤖ᯝࢆᴫࡡ෌⌧࡛ࡁࡿࡇ࡜ࢆ☜ㄆࡋࡓࠋᅗ㸲 ඹ᣺᭤⥺㸦Case1㸸㸱ᒙ㸧121

ᅗ㸳 ඹ᣺᭤⥺㸦Case2㸸㸱ᒙ㸧ᅗ㸴 ඹ᣺᭤⥺㸦Case5㸸㸱ᒙ㸧122௒ᚋࡣ㸪ࢲࣥࣃ࣮ࡀ㸯ᒙ࡜㸰ᒙ࡟㓄⨨ࡉࢀࡓሙྜࡢゎᯒࣉࣟࢢ࣒ࣛࢆᵓ⠏ࡍࡿࡓࡵ࡟㸪
Step1㸸㉁㔞ຠᯝࡀ௜ຍࡉࢀࡓ࣐ࢺࣜࢡࢫࡢຊᏛࣔࢹࣝࡢ᳨ウ
Step2㸸ࡼࡾ⢭ᗘࡢ㧗࠸ᐇ㦂್࠿ࡽࡢ⟶ෆᦶ᧿ಀᩘࡢ⟬ฟ࡞࡝ࢆ᳨ウㄢ㢟࡜ࡋ㸪ࢲࣥࣃ࣮ࡢྲྀࡾ௜ࡁ≧ἣࢆኚ࠼ࡓゎᯒࢆྍ⬟࡜ࡍࡿ࡜ඹ࡟㸪ゎᯒ⢭ᗘྥୖࢆᅗࡿࡇ࡜ࢆ┠ᣦࡍࠋཧ⪃ᩥ ⊩ 1㸧 ⏿୰཭㸪⯪ᮌᑦᕫ㸪⸨⏣ᬛᕭ㸸័ᛶ㉁㔞ຠᯝࢆ᭷ࡍࡿᾮὶࢲࣥࣃ࣮ࢆ⏝࠸ࡓከᒙ㦵⤌ࡢ᣺ືᛂ⟅≉ᛶ㸦ࡑࡢ 1㸧ヨ㦂యᴫせ࡜᣺ືྎຍ᣺ヨ㦂㸪᪥ᮏᘓ⠏Ꮫ఍኱఍Ꮫ⾡ㅮ₇᱾ᴫ㞟㸪
pp.863-864㸪2020.9
2㸧 ⏿୰཭㸪⯪ᮌᑦᕫ㸸័ᛶ㉁㔞ຠᯝࢆ᭷ࡍࡿᾮὶࢲࣥࣃ࣮ࢆ⏝࠸ࡓ 1 ᒙไ᣺ࣇ࣮࣒ࣞࡢ᣺ືᛂ⟅≉ᛶ㸦ࡑࡢ 4㸧ᆅ㟈ᛂ⟅ゎᯒࡢ⢭ᗘ᳨ド㸪᪥ᮏᘓ⠏Ꮫ఍኱఍Ꮫ⾡ㅮ₇᱾ᴫ㞟㸪21446㸪
pp.891-892㸪2019.9
3㸧 ⏿୰཭㸸័ᛶ㉁㔞ຠᯝࢆ᭷ࡍࡿᾮὶࢲࣥࣃ࣮ࢆ⏝࠸ࡓไ᣺㦵⤌ࡢ᣺ືᛂ⟅≉ᛶ㸪㸪ᮾ໭ᕤᴗ኱Ꮫ༤ኈᏛ఩ㄽᩥ㸪pp.8,12 2017-03