Bulletin of the JSME᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟

Vol.83, No.854, 2017

Transactions of the JSME (in Japanese)㍯㍈ࡢ⺬⾜ືࡢ㈇ࡢῶ⾶࡟㛵ࡍࡿ⌮ㄽⓗ⪃ᐹ㓇஭ ⱥᶞ*1
A theoretical consideration to negative damping of hunting of wheelset
Hideki SAKAI *1*1

Kindai Univ. Dept. of Robotics Engineering
1 Umenobe, Takaya, Higashihiroshima-shi, Hiroshima 739-2116, Japan

Received: 20 February 2017; Revised: 22 May 2017; Accepted: 28 July 2017
Abstract
This paper considers ways to increase stability of wheelset hunting. The aim of this study is to introduce formulas which
represent the damped natural frequency and its damping. This has been done by analytical and not by numerical calculation.
These formulas are functions of mass, creep coefficient and other wheelset parameters. These formulas are obtained with a 2
degree of freedom wheelset model whose radius of gyration of yaw equals to a half of gauge and whose creep coefficient isisotropic, in the first part of this paper. In the second part, based on one of these formulas, ways to increase damping ofwheelset hunting are discussed. First, it is shown that a longer wavelength of geometrical hunting is effective in reducing
hunting at all vehicle speeds. Secondly, it is shown that at speeds slower than a certain speed, the damping is increased when
the ratio of the wheelset mass to the creep coefficient is larger. Further, a symbolic formula for this boundary speed is
obtained. Thirdly, if it is assumed that the wheelset mass is proportional to the square of the wheel radius, then at speeds
higher than a certain speed, the damping is increased when the wheel radius is larger. Moreover a symbolic formula for thisboundary speed is also obtained.

Keywords : Wheelset, Track, Hunting, Damping, Analytical solution, Formula ⥴   ゝ㕲㐨㌴୧࡟⺬⾜ືࡀ⏕ࡌࡿ࡜㸪஌ᚰᆅࡀᝏ໬ࡋࡓࡾ㸪ሙྜ࡟ࡼࡗ࡚ࡣ⬺⥺ࡢ༴㝤ࡀ࠶ࡿ㸬⺬⾜ືࢆ⏕ࡌ࡟ࡃࡃࡍࡿࡓࡵ࡟㸪㍯㍈ࢆ㸪ྎ㌴ᯟ࡟ᑐࡋ࡚࠶ࡿ≉ᐃࡢ⠊ᅖࡢ๓ᚋ࣭ᕥྑࡢ๛ᛶ࡛ᨭᣢࡍࡿࡀ㸪ࡑࢀ࡛ࡶ㧗㏿࡟࡞ࡿ࡜⺬⾜ືࡀ⏕ࡌࡿ㸦᪥ᮏᶵᲔᏛ఍⦅㸪2008㸧㸬⺬⾜ືࡀ⏕ࡌࡿཎᅉࡣ㸪㌴㍯㋃㠃ࡢ໙㓄࡟㉳ᅉࡍࡿ㸦ᘯὠ௚㸪1984㸧ࡢ࡛㸪㍯㍈༢యࡢ⺬⾜ືࡀ⏕ࡌ࡟ࡃ࠸࡯࡝㸪㌴୧⣔ࡢ⺬⾜ືࡶ⏕ࡌ࡟ࡃ࠸ഴྥࡀ࠶ࡿ࡜ᛮࢃࢀࡿ㸬ࡑࡇ࡛ᮏㄽᩥࡣ㸪㌴୧⣔ࡢ⺬⾜ືࡢᢚไࡢࡓࡵ࡟㸪㍯㍈༢యࡢ⺬⾜ືࡢపῶἲࢆ⪃ᐹࡍࡿ㸬㍯㍈ࡢ⺬⾜ືࡢ࠺ࡕ㸪ᗄఱᏛⓗ⺬⾜ືࡣ࿘▱࡛࠶ࡿ࠿ࡽ㸪ᮏㄽᩥ࡛ࡣືⓗ⺬⾜ື࡜ࡋ࡚㸪㍯㍈༢యࡢ⺬⾜ືࢆ⾲ࡍ≉ᛶ᰿ࡢ㈇ࡢῶ
⾶ࡢῶᑡἲ㸦ࡍ࡞ࢃࡕῶ⾶ࡢቑຍἲ㸧ࢆ⪃ᐹࡍࡿࡶࡢ࡛࠶ࡿ㸬ࡑࡢࡓࡵ㸪ᮏㄽᩥ࡛ࡣ㍯㍈༢యࡢ⺬⾜ືࢆ௨ᚋ㸪༢࡟⺬⾜ື࡜グࡍ㸬ᮏㄽᩥࡢ๓༙㒊࡛ࡣ㸪ྛタィㅖඖ࡟ࡼࡿ⺬⾜ືࡢᢚไ࡟㛵ࡍࡿぢ㏻ࡋࢆᚓࡿࡓࡵ࡟㸪᭱㐺ᨭᣢ๛ᛶࢆ⾲ࡍᩥᏐᘧࡢ㛵ᩘ㸦㡲⏣௚㸪1986㸧࡟࡞ࡽࡗ࡚㸪⺬⾜ືࡢῶ⾶࡜Ἴ㛗ࢆᩥᏐᘧ࡛⾲ࡍ㸬ࡇࡢᩥᏐᘧࢆᇶ
࡟㸪ᚋ༙㒊࡛ࡣ㸪ῶ⾶ࢆቑຍࡉࡏࡿ᪉⟇ࢆ⪃ᐹࡍࡿ㸬
 ㍯㍈ࡢᇶᮏⓗᛶ㉁
࣭ ࣔࢹࣝᅗ 1 ࡟♧ࡉࢀࡿ㍯㍈ࡢࣔࢹࣝࢆ⏝࠸ࡿ㸬㌶㐨ࡣ┤⥺࡜ࡋ㸪㌶㐨ࡢ୰ᚰ⥺࡜㌴㍈୰ᚰ⥺࡜ࡢ஺Ⅼ࡟ཎⅬ o ࢆタᐃࡍࡿ㸬㌶㐨ࡢ᪉ྥ࡟ x ㍈ࢆ㸪ᯖᮌ᪉ྥ࡟ y ㍈ࢆ㸪㍯㍈ࡀᕥ᪕ᅇࡍࡿ᪉ྥ࡟ṇࡢ࢔ࢱࢵࢡゅ ȥ ࢆᐃࡵࡿ㸬ࡲࡓ㍯㍈
No.17-00064 [DOI:10.1299/transjsme.17-00064], J-STAGE Advance Publication date : 14 August, 2017ᮏㄽᩥࡣ㸪➨25ᅇ஺㏻࣭≀ὶ㒊㛛኱఍ ㅮ₇ㄽᩥ㞟(2016), No.1101ࡢᥖ㍕ෆᐜ࡟ᇶ࡙࠸ࡓㄽᩥ࡛࠶ࡿ㸬*1ṇဨ㸪㏆␥኱ᏛᕤᏛ㒊㸦ࠛ739-2116 ᗈᓥ┴ᮾᗈᓥᕷ㧗ᒇ࠺ࡵࡢ㎶ 1㸧
E-mail of corresponding author: sakai@hiro.kindai.ac.jp[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 83 －

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Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)
ࡢ x ㍈᪉ྥࡢ㏿ᗘࢆ V ࡜ࡍࡿ㸬࡞࠾ ψ ࡣᚤᑠ࡜ࡋ㸪sinψҸtanψҸψ ࠾ࡼࡧ cosψҸ1 ࡀᡂࡾ❧ࡘࡶࡢ࡜ࡍࡿ㸬
γ

y

ψ
o

V

x


Fig. 1 Model of wheelset and trackḟ࡟㸪㍯㍈ࡢᛶ㉁ࢆ⾲ࡍグྕ࡜ࡋ࡚㸪㍯㍈ࡢ㉁㔞ࢆ m㸪y=0 ࠿ࡘ ψ=0 ࡢ࡜ࡁࡢ㌴㍯ࡢ㌿ࡀࡾᅇ㌿༙ᚄࢆ r0㸪㋃
㠃໙㓄ࢆ γ㸪࣮ࣚ័ᛶ༙ᚄࢆ i㸪࣮ࣞࣝ࡜㌴㍯࡜ࡢ᥋ゐⅬ࠿ࡽ㌶㐨ࡢ୰ᚰࡲ࡛ࡢ㊥㞳ࢆ b㸪๓ᚋࢡ࣮ࣜࣉಀᩘࢆ κ11㸪ᕥྑࢡ࣮ࣜࣉಀᩘࢆ κ22 ࡜グࡍ㸬ࡇࡢࣔࢹࣝࡢ୪㐍࠾ࡼࡧᅇ㌿࡟ࡘ࠸࡚ࡢ㐠ື᪉⛬ᘧࡣࡑࢀࡒࢀḟᘧ࡛୚࠼ࡽࢀࡿ㸦᪥ᮏᶵᲔᏛ఍⦅㸪2008㸧
㸬

my



2N 22
y  2N 22\
V

mi 2\

(1)

2N11b 22N bJ
\  11 y
V
r0

(2)

ᘧ(1)㸪(2)ࡢึᮇ᮲௳ࢆ඲࡚ 0 ࡜ࡋ࡚ࣛࣉࣛࢫኚ᥮ࢆ࠾ࡇ࡞࠸㸪ࡑࡢ≉ᛶ᪉⛬ᘧࢆồࡵࡿ࡜ḟࡢࡼ࠺࡟࡞ࡿ㸬s4 

2 b 2N11  i 2N 22 3 4b 2N11N 22 2 4bN11N 22Js  2 2 2 s  2 2
i 2 mVi mVi m r0(3)0ࡇࡇ࡛ s ࡣࣛࣉࣛࢫ₇⟬Ꮚ࡛࠶ࡿ㸬
ᘧ(3)࡟♧ࡉࢀࡿࡼ࠺࡟㸪㍯㍈ࡢ⺬⾜ືࢆ⾲ࡍ≉ᛶ᪉⛬ᘧࡣ s ࡢ 4 ḟᘧ࡛࠶ࡿ㸬࣭ ≉ᛶ᪉⛬ᘧࡀ V ࡢ  ḟᘧ௨ୖࡢሙྜࡢゎᯒἲ ࡶࡋࡶ≉ᛶ᪉⛬ᘧࡀ s ࡢ 2 ḟᘧ࡛࠶ࢀࡤ㸪ᅛ᭷᣺ືᩘࡸῶ⾶ẚࡀᐃᘧ໬࡛ࡁࡿࡢ࡛㸪⺬⾜ື࡟ࡘ࠸࡚ࡢぢ㏻ࡋࢆᩥᏐᘧ࡛ᚓࡿࡇ࡜ࡀ࡛ࡁࡿ㸬ࡋ࠿ࡋᘧ(3)ࡣ s ࡢ 4 ḟᘧ࡛࠶ࡿ㸬ࡇࡢࡼ࠺࡞ሙྜࡢゎᯒᡭἲࡣḟࡢ 3 ࡘࡀ࠶ࡿ࡜
ᛮࢃࢀࡿ㸬
1)

Ᏻᐃ㡿ᇦ࡜୙Ᏻᐃ㡿ᇦ࡜ࡢ⮫⏺᮲௳ࢆᐃᘧ໬ࡋ㸪ࡑࡢᘧࢆ⪃ᐹࡍࡿ㸬2)s ࡢ 4 ḟᘧࢆ㸪 s ࡢ 2 ḟᘧ࡟㏆ఝࡋ࡚㸪ᅛ᭷᣺ືᩘࡸῶ⾶ẚࢆ㏆ఝⓗ࡟ᐃᘧ໬ࡍࡿ㸬3)s ࡢ 4 ḟᘧࢆᅉᩘศゎࡋ࡚㸪2 ࡘࡢ s ࡢ 2 ḟᘧ࡟ศゎࡋ㸪ࡑࢀࡒࢀࡢᅛ᭷᣺ືᩘࡸῶ⾶ẚࢆᐃᘧ໬ࡍࡿ㸬ୖグ 1㸧ࡢ౛࡟㸪㟼Ṇ⣔࠿ࡽᙎᛶⓗ࡟ᨭᣢࡉࢀ࡚࠸ࡿሙྜࡢ㍯㍈ࡢ⺬⾜ື࡟࠾ࡅࡿ⮫⏺㏿ᗘࡢ⪃ᐹ㸦㡲⏣௚㸪

1986㸧ࡀ࠶ࡿ㸬ࡇࡢ᪉ἲࡣᮏㄽᩥ࡟ࡣ㐺ࡉ࡞࠸ࡇ࡜ࢆ 2࣭4 ⠇࡛㏙࡭ࡿ㸬ୖグ 2㸧ࡢ୍౛ࡀ㸪ḟ⠇࡛㏙࡭ࡿᗄఱᏛⓗ⺬⾜ື㸦᪥ᮏᶵᲔᏛ఍⦅㸪2008㸧࡛࠶ࡿ㸬ᗄఱᏛⓗ⺬⾜ືࡢሙྜ㸪ῶ⾶ࡀ 0 ࡛࠶ࡿ࠿ࡽ㸪ࡇࡢ᪉ἲࡶᮏㄽᩥ࡟ࡣ㐺ࡉ࡞࠸㸬ࡑࡇ࡛ᮏㄽᩥࡀ⏝࠸ࡿࡢࡀୖグ 3㸧࡛࠶ࡿ㸬 3㸧ࡢ౛ࡢ᪤ ◊✲࡜ࡋ࡚㸪⮬ື㌴࡟࠾࠸࡚ࣁࣥࢻࣝ࠿ࡽᡭࢆᨺ
ࡍሙྜࡢ㐠ືゎᯒࡀ࠶ࡿ㸬ᅗ 1 ࡟♧ࡉࢀࡿ㍯㍈ࢆ㸪⯦ࢆ୰❧఩⨨࡛ᅛᐃࡋࡓሙྜࡢ⮬ື㌴࡜ࡳ࡞ࡍ࡜㸪ࡑࡢ⮬⏤㸬 㸬ᗘࡣ ψ ࡜ y ࡢ 2 ⮬⏤ᗘ࡞ࡢ࡛㸪⮬ື㌴ࡢ㐠ື᪉⛬ᘧࡶᘧ(1),(2)࡟‽ࡌࡓᙧᘧ࡟࡞ࡿ㸬ࡓࡔࡋ㸪⮬ື㌴࡟ࡣ ψ ࡜ yࡢ㡯ࡀ࡞࠸ࡢ࡛㸪ࡑࡢ≉ᛶ᪉⛬ᘧࡣ s ࡢ 2 ḟᘧ࡛࠶ࡿ㸬ࡇࡢ≧ែ࠿ࡽ㸪ࣁࣥࢻࣝ࠿ࡽᡭࢆᨺࡋࡓሙྜࡣ㸪᧯⯦⣔ࡢ⮬⏤ᗘࡶຍ࿡ࡉࢀࡿࡢ࡛㸪ࡑࡢ≉ᛶ᪉⛬ᘧࡣ s ࡢ 4 ḟᘧ࡟࡞ࡿ㸬ࡇࡢ 4 ḟᘧࡀ≉Ṧ࡞ሙྜ࡟ཝᐦ࡟ᅉᩘศゎࡉࢀࡓࡾ㸦㓇஭㸪2015a㸧㸪୍⯡ⓗ࡞ሙྜ࡟㏆ఝⓗ࡟ᅉᩘศゎࡉࢀࡓ㸦㓇஭㸪2015b㸧ࡢࡀ 3㸧ࡢ౛࡛࠶ࡿ㸬[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 84 －

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Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)࣭ ᗄఱᏛⓗ⺬⾜ືࡢᅛ᭷᣺ືᩘ
๓⠇ 2)ࡢ᪉ἲ࡜ࡋ࡚㸪ᘧ(3)ࡢグྕゎࢆồࡵࡿᚑ᮶ࡢ᪉ἲ㸦᪥ᮏᶵᲔᏛ఍⦅㸪2008㸧࡟ࡘ࠸࡚㏙࡭ࡿ㸬ࡲࡎ㸪
(4)

m 0

࡜௬ᐃࡍࡿ㸬ᘧ(3)ࡢ୧㎶࡟ m2 ࢆ஌ࡌࡓୖ࡛㸪ᘧ(4)ࢆ௦ධࡍࡿࡇ࡜࡟ࡼࡾ㸪ᘧ(3)ࡣ㸪
b 2 J
s 
r0
V2

(5)

0

࡜࡞ࡿ㸬ࡋࡓࡀࡗ࡚ m=0 ࡢሙྜࡢᅛ᭷᣺ືᩘࢆ ωst ࡜グࡍ࡜Z st

V 2J   br0  (6)࡜࡞ࡿ㸬ࡇࢀࡀᗄఱᏛⓗ⺬⾜ືࡢᅛ᭷᣺ືᩘ࡛࠶ࡿ㸬ࡲࡓ㸪ωst ࢆἼ㛗 S1 ࡛⾲ࡍ࡜㸪

S1

V

2S

Z st

2S

br0 

  (7)J࡜࡞ࡿ㸦᪥ᮏᶵᲔᏛ఍⦅㸪2008㸧㸬ᘧ(5)࡟࠾࠸࡚ῶ⾶ẚࡣ 0 ࡔ࠿ࡽ㸪2㸧ࡢ᪉ἲࡣ㸪ᮏㄽᩥ࡟ࡣ㐺ࡉ࡞࠸࡜ᛮࢃࢀ
ࡿ㸬࣭ ⺬⾜ືࡢᏳᐃᛶ
2࣭2 ⠇ࡢ 1)ࡢ᪉ἲ࡜ࡋ࡚㸪ᮏ⠇࡛ࡣ Huruwitz ࡢᏳᐃุู㸦᪂㸪1996㸧ࢆᘧ(3)࡟ࡘ࠸࡚⾜࡞࠸㸪Ᏻᐃ࡜୙Ᏻᐃ
࡜ࡢ⮫⏺᮲௳ࡢ⪃ᐹࢆヨࡳࡿ㸬ᘧ(3)ࡀᏳᐃ࡛࠶ࡿࡓࡵ࡟ࡣ㸪ḟࡢ 2 ࡘࡢ᮲௳ࡀྠ᫬࡟‶ࡓࡉࢀࡿᚲせࡀ࠶ࡿ㸬1 ࡘ┠ࡢ᮲௳ࡣ㸪ᘧ(3)ࡢ s ࡢಀᩘࡢ➢ྕࡀࡍ࡭࡚➼ࡋ࠸ࡇ࡜࡛࠶ࡿ㸬ᘧ(3)ࡢሙྜ㸪γ ࡀṇ࡞ࡽࡤ㸪ࡍ࡭࡚ࡢಀᩘࡀṇ࡟࡞ࡿࡢ࡛㸪ࡇࡢ᮲௳ࡣ‶ࡓࡉࢀ࡚࠸ࡿ㸦γ ࡀ㈇࡞ࡽࡤ㸪ࡇࡢ᮲௳ࡀ‶ࡓࡉࢀ࡞࠸ࡢ࡛㍯㍈ࡣᖖ࡟୙Ᏻᐃ࡛࠶ࡿ㸧㸬ࡶ࠺ 1 ࡘࡢ᮲௳ࡣ Huruwitz
ࡢ㤳ᗙᑠ⾜ิᘧࡢ➢ྕ࡟ࡘ࠸࡚࡛࠶ࡿ㸬ᘧ(3)ࡢሙྜ㸪s4 ࡢಀᩘࡀṇ࡞ࡢ࡛㸪ḟࡢ 2 ࡘࡢ㤳ᗙᑠ⾜ิᘧࡢ್ࡀࡍ࡭
࡚ṇ࡛࠶ࡿࡇ࡜ࡀᏳᐃᛶࢆ‶ࡓࡍࡓࡵࡢ 2 ࡘ┠ࡢ᮲௳࡛࠶ࡿ㸬
2 b 2N 11  i 2N 22
i 2 mV1

2 b 2N 11  i 2N 22
i 2 mV10

0
4b 2N 11N 22
i 2 m 2V 2

2 b 2N 11  i 2N 22 4b 2N 11N 22!0i 2 mVi 2 m 2V 200

4b 2N 11N 22
i 2 m 2V 2
2 b 2N 11  i 2N 22
i 2 mV4bN 11N 22J
i 2 m 2 r0(8)

2

ª 2 b 2N 11  i 2N 22 º 4bN 11N 22J
«
0
»
2
2
i 2 mV¬¼ i m r0

(9)0

[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 85 －

3

Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)γ ࡀṇ࡞ࡽࡤ㸪ᘧ(9)ࡣᖖ࡟㈇࡛࠶ࡿ࠿ࡽ㸪2 ࡘ┠ࡢ᮲௳ࡣᖖ࡟‶ࡓࡉࢀ࡞࠸㸬ࡼࡗ࡚㸪㍯㍈ࡣᖖ࡟୙Ᏻᐃ࡛࠶ࡿ㸬ࡑࡢࡓࡵ㸪Ᏻᐃ࡜୙Ᏻᐃ࡜ࡢ⮫⏺᮲௳ࡀᏑᅾࡋ࡞࠸㸬ࡋࡓࡀࡗ࡚㸪 1㸧ࡢ᪉ἲࡶ㸪ᮏㄽᩥ࡟ࡣ㐺ࡉ࡞࠸㸬 ⺬⾜ືࡢῶ⾶2࣭2 ⠇ࡢ 1㸧ࡢ᪉ἲࡶ 2㸧ࡢ᪉ἲࡶ㍯㍈༢యࡢ⺬⾜ືࡢῶ⾶ࡢ⪃ᐹ࡟ࡣ㐺ࡉ࡞࠸࡜ᛮࢃࢀࡿ㸬ࡑࡇ࡛ᮏㄽᩥ࡛
ࡣ 3㸧ࡢ᪉ἲ࡜ࡋ࡚㸪ᘧ(3)ࢆ㏆ఝⓗ࡟ᅉᩘศゎࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪ῶ⾶ࢆᐃᘧ໬ࡍࡿ㸬࣭ ௬ᐃ
ᮏㄽᩥ࡛ࡣ㸪ᘧ(3)ࢆ㏆ఝⓗ࡟ゎࡃࡓࡵࡢ௬ᐃ࡜ࡋ࡚㸪
i

(10)



b

࠿ࡘ

N11 N 22 N

(11)



࡜ࡍࡿ㸬ᘧ(4)ࡢ௬ᐃࡣ⏝࠸࡞࠸㸬࣭ ≉ᛶ᪉⛬ᘧ
ᘧ(10)㹼(11)ࢆᘧ(3)࡟௦ධࡍࡿ࡜

2§ 2N · 3 § 2N · 2 JV § 2N ·s 4  2¨¨¸¸ s ¸s  ¨r0 b © mV ¹
© mV ¹© mV ¹
2

2

(12)

0

࡜࡞ࡿ㸬ࡇࡇ࠿ࡽࡋࡤࡽࡃࡣ㸪⾲グࢆ⡆༢࡟ࡍࡿࡓࡵ࡟

K mV {

2N                                       
mV

(13)

࡜୍᫬ⓗ࡟࠾ࡃ㸬ᘧ(13)ࢆᘧ(12)࡟௦ධࡍࡿ࡜

s 4  2K mV s 3  K mV s 2  Zst K mV
2

2

2

(14)

0

࡜࡞ࡿ㸬࡞࠾㸪KmV ࡢḟඖࡣ ωst ࡢࡑࢀ࡜ྠᵝ㸪ゅ㏿ᗘ࡜⪃࠼ࡿࡇ࡜ࡀ࡛ࡁࡿ㸬࣭ ≉ᛶ᪉⛬ᘧࡢ」஧ḟᘧ໬࡜ࡑࡢゎ
ᘧ(14)ࢆᩘ್ⓗ࡟ゎ࠸ࡓ౛ࢆ⾲ 1 ࡟♧ࡍ㸬⾲ 1 ࡟♧ࡉࢀࡿࡼ࠺࡟㸪௒ᅇࡢㅖඖ࡛ࡣ㸪஧⤌ࡢ᰿ࡢ⹫㒊ࡢ⤯ᑐ್ࡀ஫࠸࡟➼ࡋ࠿ࡗࡓ㸬ࡑࡇ࡛ࠕ஧⤌ࡢ᰿ࡢ⹫㒊ࡢ⤯ᑐ್ࡀ஫࠸࡟➼ࡋ࠸ࠖ࡜௬ᐃࡍࡿ࡜㸪ᘧ(14)ࡢ஧⤌ࡢ᰿ࡣḟࡢࡼ࠺࡟᭩ࡅࡿ㸬sa r bj(15)sc r bj(16)[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 86 －

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Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)ࡇࡇ࡛㸪a㸪b㸪c ࡣࡑࢀࡒࢀᐇᩘ࡛࠶ࡾ㸪j ࡣ⹫ᩘ༢఩࡛࠶ࡿ㸬ᘧ(15)㸪(16)࡟グࡉࢀࡿ 4 ᰿ࢆゎ࡟ᣢࡘኚᩘ sࡢ᪉⛬ᘧࡣ㸪ḟࡢࡼ࠺࡟᭩ࡅࡿ㸬( s  a  bj )(s  a  bj )(s  c  bj )(s  c  bj )(17)s 4  2(c  a) s 3  (a 2  2b 2  c 2  4ca ) s 2  2(c  a)(b 2  ca ) s  (a 2  b 2 )(b 2  c 2 )0ᘧ(17)ࡢ s3 ࠾ࡼࡧ s1 ࡢಀᩘ࡟ࡣ 2(c+a)ࡢ㡯ࡀ࠶ࡿ㸬ࡋࡓࡀࡗ࡚㸪ࡇࡢ㡯ࡀ 0 ࡟࡞ࡿࡼ࠺࡞᪂ࡓ࡞ኚᩘ s’࡛ s ࢆ⨨ࡁ᥮࠼ࡿ࡜㸪s’3 ࠾ࡼࡧ s’1 ࡢಀᩘࡀྠ᫬࡟ 0 ࡟࡞ࡿࡢ࡛㸪ᘧ(17)ࡣ s’࡟ࡘ࠸࡚ࡢ」஧ḟᘧ࡟࡞ࡿ㸬ࡇࡢ」஧ḟᘧࢆ஧ḟ᪉⛬ᘧࡢゎࡢබᘧࢆ⏝࠸࡚ゎ࠸ࡓᚋ࡛㸪s ࡟෌ኚ᥮ࡍࡿࡇ࡜࡟ࡼࡗ࡚ s ࡢグྕゎࡀồࡵࡽࢀࡿࡣࡎ࡛࠶ࡿ㸬ࡑࡇ࡛㸪ࡇࡢኚᩘኚ᥮ࢆ⾜࠺㸬2(c+a)ࡣ s ࡢ 4 ᰿ࡢᐇ㒊ࡢ⥲࿴࡜➼ࡋ࠸࠿ࡽ㸪4 ᰿ࡢᐇ㒊ࡢ⥲࿴ࡀ 0 ࡟࡞ࡿࡼ࠺
࡟ s’ࢆᐃࡵࢀࡤⰋ࠸㸬ࡼࡗ࡚㸪s ࡢ 4 ᰿ࡢᐇ㒊ࡢ⥲࿴ 2(c+a)ࢆ 4 ➼ศࡋࡓ 2(c+a)/4 ࢆ s ࠿ࡽᘬ࠸ࡓࡶࡢࡀ s’࡟࡞ࡿࡣࡎ࡛࠶ࡿ㸬ࡋࡓࡀࡗ࡚ 4 ࡘࡢ s’ࡣ㸪ᘧ(15)㸪(16)࠿ࡽ㸪s' { s 

2(a  c)42(a  c) º
ª
r bj
«¬a 
4 »¼

(18)

s' { s 

2(a  c)42(a  c) º
ª
r bj
«¬c 
4 »¼

(19)

࡜᭩ࡅࡿ㸬ࡇࡇ࡛-2(c+a)ࡣ㸪ᘧ(17)࡜ᘧ(14)ࡢࡑࢀࡒࢀࡢ s3 ࡢಀᩘ࡝࠺ࡋࢆẚ㍑ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪 2(c  a) 2K mV(20)

࡛࠶ࡿ㸬ࡇࡢᘧࢆ㸪ᘧ(18)ࡲࡓࡣᘧ(19)࡟௦ධࡍࡿ࡜

s' s 

2 K mV4(21)࡜࡞ࡿ㸬ࡇࡢ s’ࢆᘧ(14)࡟௦ධࡋ㸪ᩚ⌮ࡍࡿ࡜㸪s' 4 

2
2·
K mV 2 § 2
K2s' ¨¨ Z st K mV  mV ¸¸
2
16 ¹
©

0                          (22)࡜࡞ࡿ㸬ᘧ(22)ࡣ㸪๓㏙ࡢࡼ࠺࡟ s’2 ࡟ࡘ࠸࡚ࡢ஧ḟ᪉⛬ᘧ࡛࠶ࡿ㸦ࡼࡗ࡚ᮏ⠇➨୍ẁⴠࡢ௬ᐃࡣṇࡋ࠸㸧㸬ࡑࡇ࡛㸪ᘧ(22)ࢆ㸪஧ḟ᪉⛬ᘧࡢゎࡢබᘧࢆ⏝࠸࡚ s’2 ࡟ࡘ࠸࡚ゎࡃ࡜㸪

2

s' 2

·
Z
§ K mV · §¨¸ ¨¨1 r 4 st j ¸¸ K mV ¹© 2 ¹ ©

                         (23)࡜࡞ࡿ㸬ࡇࡢᘧࡢṇ㈇ࡢᖹ᪉᰿ࢆ࡜ࡿ࡜

[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 87 －

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Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)

s' r

Z
K mV
1 r 4 st j  
2K mV                        (24)࡜࡞ࡿ㸬ࡇࡢᘧࡢ᰿ྕෆࡣ」⣲ᩘ࡛࠶ࡿ࠿ࡽ㸪s’ࡶ」⣲ᩘ࡛࠶ࡿ㸬ࡓࡔࡋ㸪᰿ྕࡢࡓࡵ࡟㸪ᐇ㒊࡜⹫㒊࡜ࡀศ㞳ࡉࢀ࡚࠸࡞࠸㸬Table 1 Numerical example of roots of eq. (12): These calculations are conducted with the following parameters: m=1600[kg],b=i=0.6[m], κ= (7.55+6.77)/2[MN/rad], γ=1/20[rad] , r0=0.43[m]. Imaginary parts of two pairs of roots were consistent.
A pair of roots
V [m/s]Real part [1/s]

The other pair of roots

Imaginary part [1/s]Real part [1/s]Imaginary part [1/s]80-7.93873±30.8410

119.814±30.8410100-11.9170±34.7647

101.417±34.7647120-15.4763±37.3334

90.0596±37.3334࣭ ゎࡢ⹫㒊࡜ᐇ㒊࡜ࡢศ㞳ࡇࡢ⠇࡛ࡣ㸪஧ḟ᪉⛬ᘧࡢゎ࡜ಀᩘࡢ㛵ಀࢆ฼⏝ࡋ࡚㸪ᘧ(24)ࡢᐇ㒊࡜⹫㒊࡜ࢆศ㞳ࡍࡿ㸬ࡲࡎ㸪ᘧ(24)ࡢ s’ࢆḟࡢࡼ࠺࡟㆑ูࡍࡿ㸬

s'1



K mV
Z
1  4 st j
2
K mV

(25)

s' 2



K mV
Z
1  4 st j
2
K mV

(26)

s '3



K mV
Z
1  4 st j (27)
2
K mV

s' 4



K mV
Z
1  4 st j 
2
K mV

(28)

ᐇ㒊ࡀඹ࡟ṇࡢ s’1 ࡜ s’2 ࡜ࡀ୍⤌ࡢඹᙺ」⣲᰿ࢆ㸪ᐇ㒊ࡀඹ࡟㈇ࡢ s’3 ࡜ s’4 ࡜ࡀࡶ࠺୍⤌ࡢඹᙺ」⣲᰿ࢆࡑࢀࡒࢀᵓᡂࡍࡿ㸬ࡇࡇ࡛㸪s’1 ࡜ s’2 ࢆ᰿࡟ᣢࡘ᪉⛬ᘧࡣ(s's'1 )(s's'2 ) 0





(29)

    (30)ࡔ࠿ࡽ㸪ࡇࢀࢆᒎ㛤ࡍࡿ࡜s'2 (s'1  s'2 )s' s'1 s'20࡜࡞ࡿ㸬ࡇࡇ࡛ s’1 ࡜ s’2 ࡢᐇ㒊ࡀඹ࡟ṇ࡛࠶ࡿࡇ࡜࡟ὀ┠ࡍࡿ࡜㸪ୖᘧࡣ

[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 88 －

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Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)s' 2  (s'1  s' 2 ) 2 s' s'1 s' 2



0

 

(31)



࡜ኚᙧ࡛ࡁࡿ㸬ᘧ(31)࡟ᘧ(25)㸪(26)ࢆ௦ධࡍࡿ࡜

s' 2 K mV

1  1  16

2

Z st 2
K mV

s'
2

K mV
4

Z st 22

1  16

K mV

2

(32)

0

࡜࡞ࡿ㸬ࡋࡓࡀࡗ࡚㸪஧ḟ᪉⛬ᘧࡢゎࡢබᘧࡼࡾ㸪s’ࡣ

s'

K mV

1  1  16

2 2

Z st 2

K mV

K mV

2 2

r
2

1  16

Z st 2
K mV

2

(33)

1 j

࡜࡞ࡿ㸬ࡇࡢᘧࡢࡍ࡭࡚ࡢ᰿ྕෆࡣṇࡔ࠿ࡽ㸪ᘧ(33)ࡢྑ㎶➨ 1 㡯ࡣᐇᩘ㸪➨ 2 㡯ࡣ⣧⹫ᩘ࡛࠶ࡿ㸬ࡋࡓࡀࡗ࡚㸪ࡇࡢᘧࡀ㸪ᘧ(24)ࡢᐇ㒊࡜⹫㒊࡜ࢆศ㞳ࡋࡓᘧ࡛࠶ࡿ㸬࣭ ≉ᛶ᰿ࡢᐃᘧ໬
ᘧ(33)ࡢ s’ࢆ㸪ᘧ(21)ࢆ౑ࡗ࡚ s ࡟ᡠࡍ࡜
§
· §
Z 2
K ¸ ¨K¨Ks ¨ mV 1  1  16 st 2  mV ¸ r ¨ mV
2 ¸ ¨2 2
K mV
¨2 2
©
¹ ©

1  16·
Z st 2¸ 1¸ j
2
K mV
¸
¹

(34)

࡜࡞ࡿ㸬ࡇࡢᐇ㒊࡟-1 ࢆ஌ࡌࡓࡶࡢࡀ㸪↓ῶ⾶ᅛ᭷᣺ືᩘ ωn ࡜ῶ⾶ẚ ζ ࡜ࡢ✚ࡔ࠿ࡽ

]Z n

§
·
§
·
2
Z st 2¨¸¨ 1 1 4 m J V 4¸   2¨ 1  1  16
¸
2
¸
K mV
N br0
K mV ¨
N ¨
¸
¨

1 
 1¸  0
¸
mV ¨
2 ¨
2
2
¸
¨
¸
¨
¸
¨
¸
¨
¸
©
¹
©
¹

 (35)

࡜࡞ࡿ㸬୍⯡࡟ ζωn ࡀ᣺ືࡢῶ⾶ࡢ㏿ࡉࢆ⾲ࡍࡀ㸪ࡇࡢᘧࡢ್ࡣ㈇࡞ࡢ࡛㸪s’1 ࡜ s’2 ࡟⏤᮶ࡍࡿ ζωn ࡣⓎᩓࡢ㏿ࡉࢆ⾲ࡍ㸬ࡲࡓᘧ(34)ࡢ⹫㒊ࡀ㸪ῶ⾶ᅛ᭷᣺ືᩘ ωd ࢆ⾲ࡍ࠿ࡽ㸪

Zd

K mV
2 2

1  16

Z st 2
1
2
K mV

N
2mV

1 4

m2 J 4               V 1
N 2 br0      (36)࡜࡞ࡿ㸬ᘧ(36)࡟♧ࡉࢀࡿ ωd ࢆἼ㛗 S1 ࡟᥮⟬ࡍࡿ࡜

S1

2S

V

Zd

(37)

2mV 2

2S
k

1 4

m2 J 4V 1
N 2 br0[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 89 －

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Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)࡜࡞ࡿ㸬୍᪉㸪s’3 ࡜ s’4 ࡟ࡘ࠸࡚ࡶ㸪 s’3 ࡜ s’4 ࡢᐇ㒊ࡀඹ࡟㈇࡛࠶ࡿࡇ࡜࡟␃ពࡋ࡚㸪ᘧ(32)㹼(36)࡜ྠᵝ࡟≉ᛶ᰿ࡢᐇ㒊࡜⹫㒊ࢆồࡵࡿ࡜㸪ࡑࢀࡒࢀ

]Z n

Zd

·
§
Z st 2
¸
¨
1
1
16


2
¸
¨
K mV
K mV ¨
 1¸
¸
2 ¨
2
¸
¨
¸
¨
¹
©

K mV

1  16

2 2

Z st 2
1
2
K mV

·
§
2
¸
¨ 1 1 4 m J V 4
2
¸
¨
N br0
N
¨
 1¸ ! 0
2
mV ¨
¸
¸
¨
¸
¨
¹
©

N
2mV

1 4

(38)



m2 J 4V 1
N 2 br0(39)࡜࡞ࡿ㸬ᘧ(38)ࡢ್ࡣṇࡔ࠿ࡽ㸪s’3 ࡜ s’4 ࡟⏤᮶ࡍࡿ᰿ࡣᏳᐃ᰿࡛࠶ࡿ㸬ࡲࡓᘧ(39)ࡣᘧ(36)࡜ྠ್࡛࠶ࡿ㸬ࡋࡓࡀࡗ࡚㸪ᘧ(35),(36),(38)ࡀ㸪ᘧ(10),(11)ࡢ௬ᐃୗࡢ≉ᛶ᰿ࡢཝᐦゎࡢグྕゎ࡛࠶ࡿ㸬 グྕゎࡢ᳨ド࣭ ㏆ఝ⢭ᗘ㐨㎷ࡢ㈨ᩱ㸦2015㸧࡟グ㍕ࡉࢀ࡚࠸ࡿ್ࢆ⏝࠸ࡓሙྜࡢᩘ್ゎ࡜㸪κ=(κ11+κ22)/2 ࡜ࡋ࡚ᘧ(35),(36),(38)ࢆ⏝࠸࡚ィ⟬ࡋࡓ⤖ᯝ࡜ࡢẚ㍑ࢆᅗ 2 ࡟♧ࡍ㸬ࡇࡢᩘ್ࡣᅗ 2 ࡢ࢟ࣕࣉࢩࣙࣥ࡟グ㍕ࡋ࡚࠶ࡿ㸬ࡲࡓ㸪i ࡢ್ࡀ୙᫂ࡔࡗ
ࡓࡢ࡛ i=b ࡜ࡋࡓ㸬௨ᚋ㸪ࡇࡢᩘ್ࢆᇶ‽ㅖඖ࡜グࡍ㸬୧⪅ࡢẚ㍑࠿ࡽ㸪ᇶ‽ㅖඖ࡟࠾࠸࡚ࡣ㸪ᘧ(11)ࡢ௬ᐃࡢᙳ㡪ࡣ㢧ⴭ࡛࡞࠸ࡼ࠺࡟ぢ࠼ࡿ㸬࡞࠾㸪௨ᚋᮏㄽᩥ࡛
ࡣ㸪୙Ᏻᐃ᰿ࢆ⾲ࡍᘧ(35),(36)ࡔࡅ࡟ࡘ࠸࡚ᢅ࠺㸬ࡑࡢࡓࡵ㸪
ࠕ㈇ࡢῶ⾶ࡢῶᑡࠖࢆ༢࡟ࠕῶ⾶ࡢቑຍࠖ࡜௨ᚋグࡍ㸬V [m/s]80

κ11=7.55[MN], κ22=6.77[MN/rad]κ 11= κ22=(7.55+6.77)/2 [MN/rad]120
2πV/ωd [m]ζωn [1/s]

0

40

-10

-20

κ11=7.55[MN], κ22=6.77[MN/rad]κ 11= κ22=(7.55+6.77)/2 [MN/rad]

20

10

0

  



(A) Real part of unstable root

2πV/ωd [m]ζωn [1/s]

100
50

80

80

120

κ11=7.55[MN], κ22=6.77[MN/rad]κ 11= κ22=(7.55+6.77)/2 [MN/rad]150

40
V [m/s]V [m/s](B) Imaginary part of unstable root

κ11=7.55[MN], κ22=6.77[MN/rad]κ 11= κ22=(7.55+6.77)/2 [MN/rad]

0

40

20

10

0

120

40

V [m/s]80120(C) Real part of stable root(D) Imaginary part of stable root
Fig. 2 Example of calculation; These calculations are conducted with the following parameters: m=1600[kg], b=0.6[m],κ11=7.55[MN], κ22=6.77[MN/rad], γ=1/20[rad] , r0=0.43[m]. In addition, i =b. These lines were calculated with eq. (35) and(36). These dots are numerical solutions of eq. (3). From these figures, it is thought that eq. (35),(36) and (38) express
fundamentals of the hunting of wheelset.

[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 90 －

8

Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)࣭ ࢡ࣮ࣜࣉಀᩘࡢ๓ᚋ࡜ᕥྑ࡜ࡢẚࡢ㏆ఝㄗᕪᇶ‽ㅖඖ࡛ࡣ κ22/κ11=0.88 ࡛࠶ࡿࡀ㸪κ22/κ11 ࡢୗ㝈್ࡣ 0.8 ࡜ࡢᣦ᦬ࡀ࠶ࡿ㸦᣺ືᕤᏛࣁࣥࢻࣈࢵࢡ⦅㞟ጤဨ఍⦅㸪1991㸧㸬ࡑࡇ࡛㸪ࡇࡢୗ㝈್ࢆ᝿ᐃࡋ࡚㸪ᅗ 2 ࡢㅖඖ࡟࠾࠸࡚ κ11+κ22 ࡢ್ࢆ୍ᐃࡢࡲࡲ㸪κ22/κ11=0.8 ࡜ࡋ࡚ィ⟬ࡋࡓ⤖ᯝࢆᅗ 3 ࡟♧ࡍ㸬ࡇࡢሙྜࡶ㸪ᘧ (11)ࡢ௬ᐃࡢᙳ㡪ࡣ㢧ⴭ࡛࡞࠸ࡼ࠺࡟ᛮࢃࢀࡿ㸬V [m/s]80120
2πV/ωd [m]ζωn [1/s]

0

40

-10
κ22/κ11 =0.8
κ22/κ11 =1

20

10

0

 

-20

 

κ22/κ11 =0.8
κ22/κ11 =1
40

V [m/s]80120(A) Real part of unstable root(B) Imaginary part of unstable root
Fig. 3 Error due to the ratio of longitudinal creep ratio κ11 to that of lateral κ22: This calculation is conducted with the values
described in fig.2 except κ11 and κ22. In this calculation, κ11+κ22=7.55+6.77 [MN] is assumed. These lines were calculatedwith eq. (35) and (36). These dots are numerical solutions of eq. (3). In case of κ22/κ11=0.8, the approximation as κ22/κ11=1
dose not yield any significant difference.

࣭ ័ᛶ༙ᚄࡢ㏆ఝㄗᕪ
㍯㍈ࡢ࠺ࡕ㸪㌴㍯ศࡢ࣮ࣚ័ᛶ༙ᚄࡣ b ࡛࠶ࡿ㸬୍᪉㸪㌴㍈ࢆ㛗ࡉ 2b ࡢ୍ᵝ࡞Წ࡜ぢ࡞ࡍ࡜㸪ࡑࡢ័ᛶ༙ᚄࡣ b/31/2
࡛࠶ࡿ㸦ᑠ㔝ᑎ㸪▮㔝㸪2014㸧㸬ࡋࡓࡀࡗ࡚㸪㍯㍈඲యࡢ័ᛶ༙ᚄࡣ b ࡼࡾࡶᑠࡉ࠸ࡶࡢ࡜ᛮࢃࢀࡿ㸬ࡑࡇ࡛ᅗ 2 ࡢィ⟬ㅖඖ࠿ࡽ i/b ࢆ 0.95 ࡟ኚ᭦ࡋ࡚㸪ᩘ್ィ⟬࡛ồࡵࡓ್࡜㸪ᘧ(35),(36)࡟ࡼࡿ್࡜ࡢẚ㍑ࢆᅗ 4 ࡟♧ࡍ㸬ᅗ 4 ࡟♧ࡉࢀࡿࡼ࠺࡟㸪⺬⾜ືࡢᇶᮏⓗᛶ㉁ࡣᘧ(35),(36)࡟ࡼࡗ࡚⪃ᐹ࡛ࡁࡿ࡜ᛮࢃࢀࡿ㸬࡞࠾㸪κ22/κ11<1 ࡟ࡼࡿㄗᕪ࡜ i/b <1 ࢆ↓どࡋࡓㄗᕪ࡜ࡣ㸪஫࠸࡟ᡴᾘࡋ࠶࠺ഃ࡟⏕ࡌࡓ㸬V [m/s]80120
2πV/ωd [m]ζωn [1/s]

0

40

-10

i/b =0.95
i/b =1

20

10

0

-20

i/b =0.95
i/b =140

V [m/s]80120(A) Real part of unstable root(B) Imaginary part of unstable root
Fig. 4 Error due to the radius of gyration i: This calculation is conducted with the values described in fig.2 except i. Thus, in thiscalculation, b is constant. These lines were calculated with eq. (35) and (36). These dots are numerical solutions of eq. (3).
Even in the approximation as i/b=1, the fundamental properties of ζωn and ωd appear to be represented by equations (35)and (36), respectively.

࣭ グྕゎࡢᡂ❧⠊ᅖ௨ୖࡢ⤖ᯝ㸪ᘧ(35),(36),(38)ࡣ㸪ᑡ࡞ࡃ࡜ࡶ 1Ӎκ22/κ11Ӎ0.8 ࠿ࡘ 1Ӎi/bӍ0.95 ࡢ⠊ᅖ࡛㸪⺬⾜ືࡢᇶᮏⓗᛶ㉁ࢆ
⾲ࡍࡶࡢ࡜ᛮࢃࢀࡿ㸬 ⪃    ᐹ࣭ ≉ᛶ᰿ࡢ኱ᇦⓗᛶ㉁ᮏ⠇࡛ࡣ㸪V ࡟ᑐࡍࡿ ζωn ࡢ኱ᇦⓗᛶ㉁ࢆ⪃ᐹࡍࡿ㸬ࡲࡎ㸪VҸ0 ௜㏆ࡢᇶᮏⓗᛶ㉁ࢆ⪃ᐹࡍࡿࡓࡵ࡟㸪ᘧ(35)ࢆ V ࡟ࡘ࠸࡚࣐ࢡ࣮ࣟࣜࣥᒎ㛤ࡍࡿ࡜
[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 91 －

9

Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)]Z n

1 § m ·§ J · 3¸V  O(V ) 7
 ¨ ¸¨¨
2 © k ¹© br0 ¸¹(40)࡜࡞ࡿ㸬ࡋࡓࡀࡗ࡚㸪VҸ0 ௜㏆࡟࠾࠸࡚㸪ᘧ(35)ࡣୖ࡟ฝࡢ㛵ᩘ࡛࠶ࡿ㸬୍᪉㸪V ࡀ༑ศ኱ࡁ࠸㡿ᇦࡢᇶᮏⓗᛶ㉁ࢆ⪃ᐹࡍࡿࡓࡵ࡟㸪VЍ҄࡜ࡍࡿ࡜2Lim ]Z n

V of§ N · § J ·¸4 ¨ ¸ ¨¨© m ¹ © br0 ¸¹(41)࡜࡞ࡿࡢ࡛㸪ζωn ࡣ୍ᐃ್࡟཰᮰ࡍࡿ㸬ࡋࡓࡀࡗ࡚㸪࠶ࡿ V ௨ୖ࡛ࡣ㸪ᘧ(35)ࡣୗ࡟ฝࡢ㛵ᩘ࡛࠶ࡿ㸬ࡇࢀࡽࡢୖ
࡟ฝ࡜ୗ࡟ฝ࡜ࡢᙳ㡪ࡀ஫࠸࡟ᡴࡕᾘࡋྜ࠺ࡢࡀ㸪ᅗ 2(A)㹼4(A)࡟࠾ࡅࡿ VҸ85[m/s]௜㏆ࡢኚ᭤Ⅼ࡛࠶ࡿ࡜ᛮࢃ
ࢀࡿ㸬ḟ࡟㸪V ࡟ᑐࡍࡿ S1 ࡢ኱ᇦⓗᛶ㉁ࢆ⪃ᐹࡍࡿ㸬S1 ࡣ㸪VҸ0 ࡢ௜㏆࡛ࡣ

S12Sª 1 § m · 2 º br0| 2S «1  ¨ ¸ V 4 »Zd¼» J¬« 2 © k ¹

(42)

V

࡛࠶ࡿ࠿ࡽୗ࡟ฝ࡛࠶ࡿ㸬ࡲࡓ㸪VЍ࡛҄ࡣ2

Lim ZdV of

4§N · J¨ ¸© m ¹ br0(43)࡜࡞ࡿࡢ࡛ୖ࡟ฝ࡟࡞ࡿ㸬ᅗ 2(B)㹼4(B)ࡣ㸪ୗ࡟ฝ࡞ࡢ࡛㸪ᘧ(42)ࡢ㡿ᇦ࡟ᑐᛂࡍࡿࡶࡢ࡜ᛮࢃࢀࡿ㸬
ᘧ(40)㹼(43)ࡣ㸪V ࢆ㝖ࡃ࡜㸪κ/m ࡜ γ/br0 ࡜ࡢ஧ࡘࡢ㡯࡟ࡼࡗ࡚ᵓᡂࡉࢀ࡚࠸ࡿ㸬ࡑࡇ࡛㸪ḟ⠇௨㝆࡛ࡣ㸪ࡇࢀࡽࡢ㡯࡟ὀ┠ࡍࡿࡇ࡜࡟ࡼࡗ࡚㸪ῶ⾶ࡢቑຍἲࢆ⪃ᐹࡍࡿ㸬࣭ ᗄఱᏛⓗ⺬⾜ືࡢ㛗Ἴ㛗໬࡟ࡼࡿῶ⾶ࡢቑຍἲ๓⠇᭱⤊ẁⴠ࡛ᣦ᦬ࡉࢀࡓ≉ᛶ᰿ࢆᵓᡂࡍࡿ 2 㡯ࡢ࠺ࡕࡢ୍ࡘࡀ γ/br0 ࡛࠶ࡿ㸬γ/br0 ࡢ㏫ᩘࡢᖹ᪉᰿࡟ 2π ࢆ஌ࡌࡓࡶࡢࡀ㸪ᗄఱᏛⓗ⺬⾜ືࡢἼ㛗࡛࠶ࡿ㸬ࡑࡇ࡛ᮏ⠇࡛ࡣ㸪ᗄఱᏛⓗ⺬⾜ືࡢἼ㛗࡟ὀ┠ࡍࡿ㸬㟼Ṇ⣔࠿ࡽᙎᛶᨭᣢࡉࢀࡓ㍯㍈ࡀ୙Ᏻᐃ࡟࡞ࡿ⮫⏺㏿ᗘࡣ㸪
㍯㍈༢యࡢᗄఱᏛⓗ⺬⾜ືἼ㛗࡟ẚ౛ࡍࡿ㸦᪥ᮏᶵᲔᏛ఍⦅㸪2008㸧㸬୍᪉㸪ᘧ(35)࠿ࡽ㸪γ/br0 ࡀᑠࡉ࠸࡯࡝ ζωn ࡣ኱ࡁࡃ࡞ࡿ㸬ࡋࡓࡀࡗ࡚㸪ᗄఱᏛⓗ⺬⾜ືࡢ㛗Ἴ㛗໬࡟ࡼࡿ⮫⏺㏿ᗘࡢቑຍࡣ㸪γ/br0 ࡢపῶ࡟ࡼࡿ ζωn ࡢቑຍ࡟㉳ᅉࡍࡿ࡜ᛮࢃࢀࡿ㸬ࡑࡇ࡛㸪ࡇࢀࢆ☜ㄆࡋࡓࡶࡢࡀ㸪ᅗ 5 ࡛࠶ࡿ㸬ࡇࡇ࡛ࡣ㸪γ ࢆῶᑡࡉࡏࡿࡇ࡜࡟ࡼࡗ࡚㸪ᅗ 5(B)࡟♧ࡉࢀࡿࡼ࠺࡟㸪⺬⾜ືࡢἼ㛗ࢆቑຍࡉࡏࡓ㸬ࡑࡢ⤖ᯝ㸪ᅗ
5(A)࠿ࡽ㸪ண᝿࡝࠾ࡾ㸪ῶ⾶ࡀቑຍࡋࡓࡇ࡜ࡀศ࠿ࡿ㸬V [m/s]80120
2πV/ωn [m]ζωn [1/s]

0

40

-10
γ =0.93㽢1/20
γ =1㽢1/20

20

10

0

-20

γ =0.93㽢1/20
γ =1㽢1/20

40

V [m/s]80120(A) Real part of unstable root(B) Imaginary part of unstable root
Fig. 5 Influence of the gradient of wheel tread γ: This calculation is conducted with the values described in fig.2 except γ. When thegradient of wheel tread γ decrease, ζωn increases at this V.
[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 92 －

10

Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)ࡇࡢῶᑡ⋡ࡣ㸪V ࡀቑຍࡍࡿ࡯࡝኱ࡁࡃ࡞ࡿࡣࡎ࡛࠶ࡿ㸬࡞ࡐ࡞ࡽ㸪ᘧ(40)࡟♧ࡋࡓࡼ࠺࡟㸪VҸ0 ࡛ࡣ㸪ζωnࡣ γ/br0 ࡢ 1 ஌࡟ẚ౛ࡍࡿࡢ࡟ᑐࡋ㸪VҸ࡛҄ࡣ㸪ᘧ(41)࠿ࡽ γ/br0 ࡢ 1/4 ஌࡟ẚ౛ࡍࡿ࠿ࡽ࡛࠶ࡿ㸬࡞࠾㸪1>γ/br0࡞ࡽࡤ㸪(γ/br0)1/4>(γ/br0)1 ࡛࠶ࡿ㸬௨ୖࡢࡼ࠺࡟㸪ᗄఱᏛⓗ⺬⾜ືࡢ㛗Ἴ㛗໬࡟ࡼࡿ⺬⾜ືࡢᢚไࡣ㸪ᙎᛶᨭᣢࡉࢀࡓ㍯㍈ࡔࡅ࡛࡞ࡃ㸪㍯㍈༢య
࡟࠾࠸࡚ࡶ᭷ຠ࡛࠶ࡿ㸬ࡓࡔࡋ୧⪅ࡢᅉᯝ㛵ಀ࡜ࡋ࡚ࡣ㸪㍯㍈༢యࡢ⺬⾜ືࡢῶ⾶ࡀቑຍࡋࡓ⤖ᯝ࡜ࡋ࡚㸪ᙎᛶᨭᣢࡉࢀࡓ㍯㍈ࡢ⺬⾜ືࡢ⮫⏺㏿ᗘࡀቑຍࡍࡿࡶࡢ࡜ᛮࢃࢀࡿ㸬࣭ ㌴㍯㉁㔞࡟ὀ┠ࡋࡓῶ⾶ࡢቑຍἲ
ᮏ⠇࡛ࡣ㍯㍈ࡢ㍍㔞໬➼࡟ࡼࡿῶ⾶ࡢቑຍࢆ᳨ウࡍࡿ㸬m ࡀኚ໬ࡍࡿ࡜㍯㔜ࡀኚ໬ࡋ㸪κ ࡶኚ໬ࡍࡿ㸬ࡇࡇ࡛κ ࡣ㍯㔜 2/3 ஌࡟ẚ౛ࡍࡿ㸦ᶓ℩㸪1971㸧࡜௬ᐃࡍࡿ㸬ࡑࡇ࡛N2                 (44)q(m  M ) 3              ࡜グࡍ㸬ࡇࡇ࡛ M ࡣ㸪(M+m)g ࡀ㍯㔜࡟࡞ࡿࡼ࠺࡟ᐃ⩏ࡉࢀࡿ㔞࡛࠶ࡿ(g ࡣ㔜ຊຍ㏿ᗘ)㸬ࡍ࡞ࢃࡕ 2 ㍈ྎ㌴ࡢሙྜࡢ M ࡣ㸪㍯㍈ࢆ㝖࠸ࡓ㌴୧㉁㔞ࡢ 1/4 ࡛࠶ࡿ㸦㐃᥋㌴ࢆ㝖ࡃ㸧㸬ࡲࡓ q ࡣẚ౛ᐃᩘ࡛࠶ࡾ㸪ᇶ‽ㅖඖ࡜ᩚྜࡍ
ࡿࡼ࠺࡟タᐃࡉࢀࡿ㸬m ࡀ ζωn ࡟ཬࡰࡍᙳ㡪ࡢ┠Ᏻ࡜ࡋ࡚㸪m ࢆᚤᑠ࡟ኚ໬ࡉࡏ࡚ࡶ㸪ζωn ࡀኚ໬ࡋ࡞࠸ V ࢆồࡵࡿ㸬ࡇࡢ V ࡟࠾࠸
࡚
w]Z n
wm

(45)

0

ࡀᡂࡾ❧ࡘᚲせࡀ࠶ࡿ㸬ࡇࡢ V ࡣ㸪ᘧ(44)ࢆ⪃៖ࡋࡓሙྜ㸪ゎᯒⓗ࡟ࡣồࡲࡽ࡞࠸㸬ᘧ(44)࡟グࡉࢀࡿ㍯㔜ࡢ 2/3
஌ࡣ㸪㍯㔜ࡢ 1 ஌࡜㍯㔜ࡢ 0 ஌࡜ࡢ㛫࡟࠶ࡿ㸬ࡑࡇ࡛୧⪅ࡢሙྜࢆᐃᘧ໬ࡍࡿ㸬ࡲࡎ๓⪅ࡢሙྜ࡜ࡋ࡚           (46)N q' ( m  M )                    ࡜௬ᐃࡍࡿ㸬ࡇࡇ࡛ q’ࡣᐃᩘ࡛࠶ࡾ㸪ᇶ‽ㅖඖ࡜ᩚྜࡍࡿࡼ࠺࡟タᐃࡉࢀࡿ㸬ୖᘧࢆᘧ(35)࡟௦ධࡋࡓࡶࡢࢆࡉ
ࡽ࡟ᘧ(45)࡟௦ධࡋ㸪ࡑࢀࢆ‶ࡓࡍ V ࢆồࡵࡿ࡜V

4§ M · br § N ·2  5 ¨1  ¸ 0 ¨ ¸
m ¹ J ©m¹
©

2

(47)

࡜࡞ࡿ㸬ࡋࡓࡀࡗ࡚ M/m ࡀᑠࡉ࠸࡯࡝㸪ࡍ࡞ࢃࡕ m ࡀ኱ࡁ࠸࡯࡝ࡇࡢ V ࡀప㏿ഃ࡟⛣ືࡍࡿ㸬୍᪉㸪㍯㔜ࡢ 0஌㸪ࡍ࡞ࢃࡕ κ ୍ᐃ࡜ࡋ࡚㸪ᘧ(35)ࢆᘧ(45)࡟௦ධࡋ㸪ࡑࢀࢆ‶ࡓࡍ V ࢆồࡵࡿ࡜V

24
br § N ·2 5 0 ¨ ¸
J ©m¹
5

2

(48)

࡜࡞ࡿ㸬ࡇࡢሙྜࡶ㸪m ࡀ኱ࡁ࠸࡯࡝ࡇࡢ V ࡀప㏿ഃ࡟⛣ືࡍࡿ㸬୧⪅ࡢ⤖ᯝ࠿ࡽ㸪κ ࡀ㍯㔜ࡢ 2/3 ஌ࡢሙྜࡶ
m ࡀ኱ࡁ࠸࡯࡝ࡇࡢ V ࡀప㏿ഃ࡟⛣ືࡍࡿࡶࡢ࡜ᛮࢃࢀࡿ㸬ࡑࡇ࡛ᘧ(47)࠾ࡼࡧᘧ(48)࡟ࡼࡗ࡚⾲ࡉࢀࡿ V ࡟ὀ┠ࡍࡿ㸬ᘧ(47)㸪(48)࡟ᇶ‽ㅖඖࢆ௦ධࡍࡿ࡜㸪࡝ࡕࡽࡶ V=144.6[m/s]࡟࡞ࡗࡓ㸬ࡑࡇ࡛㸪ࡇࡢ V ࡼࡾࡶప㏿ഃ࡛ࡢ ζωnࡢቑຍࢆ≺ࡗ࡚㸪㍯㍈ࡢ㍍㔞໬࡟ࡼࡗ࡚ κ/m ࢆቑຍࡉࡏࡓሙྜࡢィ⟬⤖ᯝࢆᅗ 6 ࡟♧ࡍ㸬ண᝿㏻ࡾ㸪ᑡ࡞ࡃ࡜ࡶ
0ӌ V ӌ120[m/s]ࡢ⠊ᅖ࡟࠾࠸࡚ ζωn ࡀቑຍࡋࡓ㸬[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 93 －

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－ 94 －

Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)
ࡇࡢ V ࡼࡾࡶప㏿ഃ࡛ࡣ㸪r0 ࡀᑠࡉ࠸᪉ࡀ㸪㧗㏿ഃ࡛ࡣ r0 ࡀ኱ࡁ࠸᪉ࡀ㸪ῶ⾶ࡀቑຍࡍࡿ㸬ࡋࡓࡀࡗ࡚㸪ࡇࡢV ࢆ㸪r0 ࡀ ζωn ࡟ཬࡰࡍᙳ㡪ࢆ⪃ᐹࡍࡿୖ࡛ࡢᣦᶆ࡜ࡍࡿ㸬ࡇࡢ V ࡣ㸪ᇶ‽ㅖඖ࡛ࡣ V=96.6[m/s]࡛࠶ࡗࡓ㸬ࡑࡇ࡛㸪ࡇࡢ V ࡼࡾࡶ㧗㏿ഃ࡟࠾ࡅࡿῶ⾶ࡢቑຍࢆ≺ࡗ࡚㸪r0ࢆᇶ‽ㅖඖࡢ 1.2 ಸࡋࡓ࡜ࡁࡢᙳ㡪ࢆᅗ 7 ࡟♧ࡍ㸬ᅗ 7 ࡢィ⟬࡟࠾࠸࡚㸪κ ࡢィ⟬࡟ࡣᘧ(44)ࢆ⏝࠸ࡓ㸬ᅗ 7 ࡟࠾࠸࡚㸪┠Ᏻ࡛࠶ࡿ V=96.6[m/s]ࡼࡾࡶప࠸ VҸ87[m/s]ࡢ㧗㏿ഃ࡛ ζωn ࡀቑຍࡋࡓ㸬ࡋࡓࡀࡗ࡚ᮏㅖඖ࡟࠾ࡅࡿ㌴㍯
ࡢ኱ᚄ໬࡟ࡼࡿ⺬⾜ࡢపῶ⟇ࡣ㸪㉁㔞࠾ࡼࡧࢡ࣮ࣜࣉಀᩘࢆ⪃៖ࡋ࡚ࡶ㸪᭷ຠ࡛࠶ࡗࡓ㸬r0=1.2㽢0.43[m], m =1.22㽢1600[kg], κ=7.50[MN/rad]r0=1 㽢0.43[m], m =1 㽢1600[kg], κ=7.00[MN/rad]V [m/s]
40

80

120

2πV/ωd [m]ζωn [1/s]

0

-10

-20

r0=1.2㽢0.43[m], m =1.22㽢1600[kg] ,κ=7.50[MN/rad]r0=1 㽢0.43[m], m =1 㽢1600[kg] ,κ=7.00[MN/rad](A) Real part of unstable root

20

10

0

40

V [m/s]

80

120



(B) Imaginary part of unstable root

Fig.7 Influence of the radius of wheel r0 related mass of the wheelset m (κ is defined as a function of m): This calculation isconducted with the values described in fig.2 except r0 and m and κ. κ is determined by eq. (44). An increase in r0 increasedζωn at V>87[m/s].

 ⤖   ゝ័ᛶ༙ᚄࡀ㌶㛫ࡢ༙ศ࡟➼ࡋࡃ࠿ࡘ๓ᚋ࡜ᕥྑࡢࢡ࣮ࣜࣉಀᩘࡀ➼ࡋ࠸࡜ࡢ௬ᐃࢆ࠾࠸ࡓ࡜ࡁࡢ㍯㍈ࡢ⺬⾜ືࡢ≉ᛶ᰿ࡢཝᐦゎࢆᐃᘧ໬ࡋࡓ㸬≉ᛶ᰿ࡣḟࡢ 2 ࡘࡢ್࡟ࡼࡗ࡚ኚ໬ࡍࡿ㸬
1 ࡘࡣࢡ࣮ࣜࣉಀᩘࢆ㍯㍈ࡢ㉁㔞࡜㌴㏿࡜ࡢ✚࡛㝖ࡋࡓࡶࡢ࡛࠶ࡿ㸬ࡋࡓࡀࡗ࡚㸪㌴㏿ࡀ୍ᐃ࡞ሙྜࡣ㸪ࠕ㸦ࢡ࣮ࣜࣉಀᩘ㸧㸭㸦㍯㍈ࡢ㉁㔞㸧
ࠖࡢ್࡟ᕥྑࡉࢀࡿ㸬ࡶ࠺ 1 ࡘࡣ㸪ᗄఱᏛⓗ⺬⾜ືࡢᅛ᭷᣺ືᩘ࡛࠶ࡿ㸬㍯㍈ࡢ㉁㔞ࡀቑຍࡍࡿ࡜㍯㔜ࡀቑຍࡍࡿࡢ࡛㸪ࢡ࣮ࣜࣉಀᩘࡶቑຍࡍࡿ㸬ࡑࡇ࡛ࢡ࣮ࣜࣉಀᩘࡀ㍯㔜ࡢ 2/3 ஌࡟ẚ౛ࡍࡿ࡜ࡋࡓሙྜࡢࠕ
㸦ࢡ࣮ࣜࣉಀᩘ㸧㸭㸦㍯㍈ࡢ㉁㔞㸧ࠖࡀῶ⾶ẚ࡟ཬࡰࡍᙳ㡪ࢆᩥᏐᘧ࡛⪃ᐹࡋࡓ㸬ࡑࡢ⤖ᯝ㸪㍯㍈ࡢ㉁㔞ࢆ㍍㔞໬ࡍࡿ࡯࡝㈇ࡢῶ⾶ࡀῶᑡ㸦ࡍ࡞ࢃࡕῶ⾶ࡀቑຍࡍࡿ㸧ࡇ࡜ࢆᣦ᦬ࡋ
ࡓ㸬ࡓࡔࡋ㸪௒ᅇࡢィ⟬ㅖඖ࡛ࡣ㌴㏿ 145[m/s]௨ୖ࡛ࡣ㸪㍍㔞໬ࡍࡿ࡯࡝ῶ⾶ࡀపୗࡍࡿ㸬㌴㍯ࡢ㌿ࡀࡾ༙ᚄࢆቑຍࡉࡏࡿ࡜㸪ᗄఱᏛⓗ⺬⾜ືࡢᅛ᭷᣺ືᩘࡀῶᑡࡍࡿ࡯࠿㸪㍯㍈ࡢ㉁㔞ࡀቑຍࡋ㸪ࡑࡢࡓࡵࢡ࣮ࣜࣉಀᩘࡶቑຍࡍࡿࡢ࡛㸪ࠕ㸦ࢡ࣮ࣜࣉಀᩘ㸧㸭㸦㍯㍈ࡢ㉁㔞㸧
ࠖࡶኚ໬ࡍࡿ㸬ࡑࡇ࡛ࡇࡢᙳ㡪ࢆຍ࿡ࡋࡓ࡜ࡁࡢ㌿ࡀࡾ༙ᚄࡀῶ⾶࡟ཬࡰࡍᙳ㡪ࢆㄪᰝࡋ㸪㌴㏿ 120[m/s]࡛ࡣ኱ᚄ໬ࡍࡿ࡯࡝ῶ⾶ࡀቑຍࡍࡿࡇ࡜ࢆᣦ᦬ࡋࡓ㸬ࡓࡔࡋ࠶ࡿ㌴㏿ࡼࡾࡶప㏿ഃ࡛ࡣᑠᚄ໬ࡍࡿ࡯࡝ῶ⾶ࡀቑຍࡍࡿ㸬ࡇࡢ⮫⏺㌴㏿ࡣ㸪௒ᅇࡢィ⟬ㅖඖ࡛ࡣ 87[m/s]࡛࠶ࡗࡓ㸬ᮏ◊✲࡛ᐃᘧ໬ࡋࡓῶ⾶࠾ࡼࡧᅛ᭷᣺ືᩘࡢᘧࡢ㏆ఝ⢭ᗘࢆ⪃ᐹࡋࡓ㸬ᑡ࡞ࡃ࡜ࡶࠕ
㸦័ᛶ༙ᚄ㸧㸭㸦㌶㛫ࡢ༙ศ㸧
ࠖࡀ 0.95㹼1 ࠿ࡘࠕ㸦ᕥྑࢡ࣮ࣜࣉಀᩘ㸧㸭㸦๓ᚋࢡ࣮ࣜࣉಀᩘ㸧ࠖࡀ 0.8㹼1 ࡢ⠊ᅖ࡛ࡣ㏆ఝㄗᕪࡀ↓ど࡛ࡁࡿ࡜⪃࠼ࡽࢀࡿ㸬ᮏ◊✲ࡣ㍯㍈༢య࡟ࡘ࠸࡚ࡢ⪃ᐹ࡛࠶ࡗࡓࡢ࡛㸪௒ᚋࡣྎ㌴࡛ࣞ࣋ࣝࡢᩥᏐᘧ࡟ࡼࡿ⪃ᐹ࡟Ⓨᒎࡉࡏ࡚࠸ࡁࡓ
࠸࡜⪃࠼࡚࠸ࡿ㸬
[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 95 －

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Sakai, Transactions of the JSME (in Japanese), Vol.83, No.854 (2017)ㅰ   ㎡ᮏ◊✲࡟࠶ࡓࡾ㸪᪥ᮏᶵᲔᏛ఍஺㏻࣭≀ὶ㒊㛛㕲㐨ᢏ⾡ጤဨ఍ࡢጤဨࡢⓙᵝ࡟ࡈᣦᑟࢆ㡬ࡁࡲࡋࡓ㸬ࡇࡇ࡟ឤㅰࡢពࢆ⾲ࡋࡲࡍ㸬ᩥ   ⊩ᘯὠဴ஧㸪᳃⏣㞝୕㸪ᒾᓮᩥ㞝㸪▼⏣࿘஧㸪㕲㐨㌴୧ࡢࡔ⾜ືࡢࢩ࣑࣮ࣗࣞࢩࣙࣥ㸦➨ 1 ሗ㸪ྛ㒊ࡢ㐟㛫ࢆ⪃៖ࡋࡓຊᏛࣔࢹࣝ࡜ࡑࡢ㐣Ώ⌧㇟㸧㸪᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟 C ⦅㸪Vol.50, No.450 (1984), pp.297-306.
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[DOI: 10.1299/transjsme.17-00064]© 2017 The Japan Society of Mechanical Engineers－ 96 －

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