近畿大学次世代基盤技術研究所報告 Vol. 7（2016）73－76

㍯㍈ࡢືⓗ࡞⺬⾜ື࡟ࡘ࠸࡚ࡢ⌮ㄽⓗ⪃ᐹ㸦ࢡ࣮ࣜࣉಀᩘ࡜㉁㔞ࢆ⪃៖ࡋࡓᅛ᭷᣺ືᩘ࡜ῶ⾶ẚࡢᐃᘧ໬㸧㓇஭ ⱥᶞ*1 

A Theoretical Study on Dynamical Weave of Wheel Set(Formulas of Natural Frequencies and Damping Ratios
Taken Account of Creep Coefficient and Mass)
Hideki SAKAI*1


In order to obtain insight about weave phenomenon of wheel set of rail vehicle, this paper
starts with formulating its natural frequencies and damping ratios. Interpreting these formulas,this paper points out that larger ratio of creep coefficient normalized by wheel set mass reducesthe weave at lower speed than a given speed and the smaller ratio does it at higher speed.
Keywords: Weave, Wheel Set, Natural Frequency, Damping Ratio㸬ࡣࡌࡵ࡟
᪂ᖿ⥺ࡢ฼౽ᛶࡢ୍ࡘࡣ㸪㏿㐩ᛶ࡛࠶ࡿ㸬㏿㐩ᛶࢆࡉࡽ࡟ྥୖࡉࡏࡿࡓࡵࡢ୍ࡘࡢ᪉ἲ࡟㸪㐠⾜᭱㧗㏿ᗘࡢቑຍࡀ࠶ࡆࡽࢀࡿ㸬ࡋ࠿ࡋ㧗㏿࡟࡞ࡿ࡟ࡘࢀ࡚㸪⺬⾜ືࡀ⏕ࡌࡸࡍࡃ࡞ࡾ㸪⺬⾜ື࡟ࡼࡗ࡚᭱㧗㏿ᗘࡀไ㝈ࡉࢀࡿ㸬⺬⾜ືࡢ࣮ࣔࢻࡢ୍ࡘ࡟㍯㍈ࡢ⺬⾜ືࡀ࠶ࡿ㸬ࡑࡢࡓ
ࡵ࡟㍯㍈ࡢ⺬⾜ື࡟ࡘ࠸࡚ࡢ◊✲ࡀ࠾ࡇ࡞ࢃࢀ࡚࠸ࡿ㸬㍯㍈ࡢ⺬⾜ືࢆ⾲ࡍ≉ᛶ᪉⛬ᘧࡣ㸪᭱ᑠ⮬⏤ᗘࡢࣔࢹࣝ
࡛ࡶᚤศ₇⟬Ꮚࡢ 4 ḟᘧ࡟࡞ࡿࡓࡵ㸪ᅛ᭷᣺ືᩘࡢᐃᘧ໬ࡀᅔ㞴࡛࠶ࡿ㸬ࡑࡇ࡛㸪ືⓗ࡞ຠᯝࢆ↓どࡍࡿࡇ࡜࡛㸪ᅛ᭷᣺ືᩘࡀᐃᘧ໬ࡉࢀࡓᗄఱᏛⓗ⺬⾜ືࡢᘧࢆ⏝࠸࡚
኱ᒁࢆぢ㏻ࡍ࠿㸪ᅛ᭷᣺ືᩘࢆᩘ್ⓗ࡟ゎࡁ㸪ࣃ࣓࣮ࣛࢱࢫࢱࢹ࢕ࢆ⾜࠺᪉ἲ࡟ࡼࡿ◊✲᪉ἲࡀ⏝࠸ࡽࢀ࡚࠸ࡿ㸬ᮏㄽᩥࡣ㸪ࡼࡾ኱ᒁࢆぢ㏻ࡋࡸࡍࡃࡍࡿࡇ࡜ࢆ┠ⓗ࡟㸪ືⓗ࡞ຠᯝࢆຍ࿡ࡋࡓሙྜࡢ⺬⾜ືࡢᅛ᭷᣺ືᩘࢆᐃᘧ
໬ࡍࡿ㸬ḟ࡟㸪ࡇࡢᘧࢆྫྷ࿡ࡍࡿࡇ࡜࡟ࡼࡾ㸪⺬⾜ືࡀ㉳ࡇࡾ࡟ࡃࡃ࡞ࡿࡓࡵࡢ᮲௳࡜ࡋ࡚㸪࠶ࡿ㌴㏿ࡼࡾప㏿ഃ࡛ࡣࢡ࣮ࣜࣉಀᩘ/㉁㔞ẚࡀ኱ࡁࡃ㸪㧗㏿ഃ࡛ࡣࢡ࣮ࣜ
ࣉಀᩘ/㉁㔞ẚࡀᑠࡉ࠸ࡇ࡜ࢆᣦ᦬ࡍࡿ㸬㸬᪤ ◊✲ ࣔ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
ࡢ x ㍈᪉ྥࡢ㏿ᗘࢆ V ࡜ࡍࡿ㸬

Ȗ

ȥoyV

ᅗ  ㍯㍈࠾ࡼࡧ㌶㐨ࡢࣔࢹࣝ࡜ࡑࡢᗙᶆ⣔ཎ✏ཷ௜ 2016 ᖺ 4 ᭶ 25 ᪥*1 ㏆␥኱ᏛᕤᏛ㒊 ࣟ࣎ࢸ࢕ࢡࢫᏛ⛉ ෸ᩍᤵ㸪ḟୡ௦ᇶ┙ᢏ⾡◊✲ᡤ ෸ᩍᤵ㸦ࠛ739-2116 ᮾᗈᓥᕷ㧗ᒇ࠺ࡵࡢ㎶ 1 ␒㸧E-mail sakai@hiro.kindai.ac.jp－ 73 －ḟ࡟㸪㍯㍈ࡢᛶ㉁ࢆ⾲ࡍグྕ࡜ࡋ࡚㸪㍯㍈ࡢ㉁㔞ࢆ m㸪
y=0 ࠿ࡘ ȥ=0 ࡢ࡜ࡁࡢ㌴㍯ࡢ㌿ࡀࡾ᪉ྥࡢᅇ㌿༙ᚄࢆ r0㸪
㋃㠃໙㓄ࢆ Ȗ㸪࣮ࣚ័ᛶ༙ᚄࢆ i㸪࣮ࣞࣝ࡜㌴㍯࡜ࡢ᥋ゐⅬ࠿ࡽ㌶㐨ࡢ୰ᚰࡲ࡛ࡢ㊥㞳ࢆ b㸪๓ᚋࢡ࣮ࣜࣉಀᩘࢆ
ț11㸪ᕥྑࢡ࣮ࣜࣉಀᩘࢆ ț22 ࡜グࡍ㸬ࡇࡢࣔࢹࣝࡢ୪㐍࠾ࡼࡧᅇ㌿࡟ࡘ࠸࡚ࡢ㐠ື᪉⛬ᘧࡣࡑࢀࡒࢀḟᘧ࡛୚࠼ࡽࢀࡿ(1)㸬
my 

2N 22y  2N 22\V

mi 2\ 

2N11b 2
2N bJ\  11 y
Vr0(1)

V 2Jy
br00Nm {2N 22m(8)(3)ࡢ㛵ಀࢆᚓࡿ㸬࡞࠾㸪ࡇࡢኚᙧ࡟㝿ࡋ(4)ࡢ㛵ಀࢆ⏝࠸ࡓ㸬ᘧ(3)࠿ࡽ㸪័ᛶ㡯ࡀ 0 ࡢሙྜࡢᅛ᭷᣺ືᩘࢆ Ȧst ࡜グࡍ࡜2V J  br0㸬័ᛶ㡯ࢆ⪃៖ࡋࡓᅛ᭷᣺ືᩘࡢᐃᘧ໬ ࣔࢹࣝᘧ(1)ࡢ୧㎶ࢆ m ࡛㸪ᘧ(2)ࡢ୧㎶ࢆ m i2 ࡛ࡑࢀࡒࢀ㝖ࡍ࡜㸪
2N 22 12Ny  22 \m Vm(6)22N m 3 N m 2 JN ms  2 s VVr0b0(9)࡜࡞ࡿ㸬ࡇࡇ࡛ s ࡣࣛࣉࣛࢫ₇⟬Ꮚ࡛࠶ࡿ㸬 ㏿ᗘ↓㝈኱ࡢሙྜࡢᅛ᭷᣺ືᩘ ㏿ᗘ↓㝈኱ࡢሙྜࡢᅛ᭷᣺ືᩘࢆ ȦVinf ࡜グࡍ࡜㸪ᘧ(9)࡟࠾࠸࡚ Vĺ࡜ࡍࡿࡇ࡜࡛

ZV inf   (5)࡜࡞ࡿ㸬ࡇࢀࡀ‽㟼ⓗ࡞ሙྜࡢ⺬⾜ືࡢᅛ᭷᣺ືᩘ࡛࠶
ࡿ㸬࡞࠾ࡇࡢሙྜ㸪ῶ⾶ẚࡣ 0 ࡛࠶ࡿ㸬y 2N11m2

y V\(7)࡜グࡍ㸬ࡉࡽ࡟኱ᒁࢆぢ㏻ࡋࡸࡍࡃࡍࡿࡓࡵ࡟ b/i=1 ࡜௬ᐃࡍࡿ㸬ḟ࡟ᘧ(6)㸪(7)࡟࠾࠸࡚㸪ึᮇ᮲௳ࡣ඲࡚ 0 ࡛࠶ࡿ࡜ࡋ࡚ࣛࣉࣛࢫኚ᥮ࢆ࠾ࡇ࡞࠸㸪ࡑࡢ≉ᛶ᪉⛬ᘧࢆồࡵࡿ࡜s4 Z st2

2N11 § b · 12N § b · Jy¨ ¸ \  11 ¨ ¸m ©i¹ Vm © i ¹ r0b࡜᭩ࡅࡿ㸬ࡑࡇ࡛኱ᒁࢆぢ㏻ࡋࡸࡍࡃࡍࡿࡓࡵ࡟㸪ț11=ț22࡜௬ᐃࡍࡿ㸬ࡇࡇ࡛ࡣ㸪⮬ື㌴࡛ࡣࢥ࣮ࢼࣜࣥࢢࢫࢸ࢕
ࣇࢿࢫ࡜࿧ࡤࢀࡿ್࡛࠶ࡾ㸪⮬ື㌴ࡢ㐠ື࡟࠾࠸࡚ࡣ㸪ࢥ࣮ࢼࣜࣥࢢࢫࢸ࢕ࣇࢿࢫ⮬యࡼࡾࡶ㸪࠶ࡿ㍯ࡢࢥ࣮ࢼࣜࣥࢢࢫࢸ࢕ࣇࢿࢫ࡜ࡑࡢ㍯ࡀ㈇ᢸࡍࡿ㉁㔞࡜ࡢẚ࡟ࡼࡾព࿡ࡀ࠶ࡿ࡜ᣦ᦬ࡉࢀ࡚࠸ࡿ(2)㸬ࡑࡇ࡛ࢡ࣮ࣜࣉಀᩘࢆ

(2) ᅛ᭷᣺ືᩘ ‽㟼ⓗ࡞ゎᯒ࡜ࡋ࡚㸪័ᛶ㡯ࢆ 0㸪ࡍ࡞ࢃࡕᘧ(1)㸪(2)ࡑࢀࡒࢀࡢᕥ㎶ࢆ 0 ࡜ࡍࡿ࡜㸪ᘧ(1)㸪(2)࠿ࡽ
y 2mi 2

4JN m 2(10)r0b࡜࡞ࡿ㸬 ୍⯡ⓗ࡞㏿ᗘࡢሙྜࡢᅛ᭷᣺ືᩘ ࡇࡢ⠇࡛ࡣᘧ(9)ࢆ㸪ኚᩘኚ᥮࡟ࡼࡗ࡚」஧ḟᘧ࡟ศゎࡍࡿ᪉ἲ(3)ࢆヨࡳࡿ㸬ࡲࡎᘧ(9)ࡢ s3 ࡢ㡯ࢆᾘཤࡍࡿࡓࡵ
࡟㸪4 ḟ᪉⛬ᘧࡢゎἲ࡛࠶ࡿ Ferrari ࡢ᪉ἲ࡟࡞ࡽࡗ࡚㸪z{sNm(11)2V࡜ࡋ࡚ᐃ⩏ࡉࢀࡿኚᩘ z ࢆᘧ(9)࡟௦ධࡍࡿ࡜㸪z4 － 74 －Nm2
V2

§ N 4 JN 2 ·z 2  ¨¨ m 4  m ¸¸ 0     r0b ¹© 16V(12)࡜࡞ࡾ㸪ண᝿㏻ࡾ z3 ࡢ㡯ࡀᾘ࠼ࡿࡔࡅ࡛࡞ࡃ㸪ண᝿እࡢࡇࢀࡣ㸪
ț11=ț22 ࡜ b/i=1ຠᯝ࡜ࡋ࡚ z1 ࡢ㡯ࡶᾘ࠼࡚ࡋࡲ࠺㸬ࢆ௬ᐃࡋࡓࡓࡵ࡛࠶ࡿ㸬௨ୖࡢ⤖ᯝ㸪ᘧ(12)ࡣ z2 ࡟ࡘ࠸࡚ࡢ஧ḟ᪉⛬ᘧ࡟࡞ࡗࡓ㸬ࡑࡇ࡛㸪ᘧ(12)ࢆ㸪ゎࡢබᘧࢆ⏝࠸࡚ z2 ࡟ࡘ࠸࡚ゎࡃ࡜ z2 ࡟ࡘ࠸࡚ࡢ஧ࡘࡢゎࡀồࡲࡾ㸪ࡉࡽ࡟ࡑࡢṇ㈇ࡢ
᰿ྕࢆ࡜ࡿ࡜ z ࡟ࡘ࠸࡚ࡢ 4 ࡘࡢゎࡀồࡲࡿ㸬ࡑࢀࡽࡢゎࢆ㸪ᘧ(11)ࡢ㛵ಀࢆ⏝࠸ࡿ࡜ s ࡟ࡘ࠸࡚ࡢ 4 ࡘࡢゎࡀồࡲࡿ㸬ࡑࢀࡽࡢゎ࡟ࡘ࠸࡚㸪஧ḟ᪉⛬ᘧࡢゎ࡜ಀᩘࡢ㛵ಀࢆ㐺⏝ࡍࡿ(3)࡜㸪ḟ࡟♧ࡉࢀࡿ஧ࡘࡢ≉ᛶ᪉⛬ᘧࡀᚓࡽࢀࡿ㸬
s 2  2] 1Z n1  Z n120(13)s 2  2] 2Z n 2  Z n 220(14)ᐃཎᅉ࡟࡜ࡋ࡚௨ᚋ㸪ᘧ(13)࡟ὀ┠ࡍࡿ㸬 ᅛ᭷᣺ືᩘ࡜ῶ⾶ẚࢆᨭ㓄ࡍࡿኚᩘ ᘧ(13)ࡣ㸪Ȧst2 ࡜ țm/(2V)ࡢ 2 ኚᩘ࡟ᨭ㓄ࡉࢀ࡚࠸ࡿ㸬ࡲࡓ㸪ࡇࢀࡽࡢ㡯ࡣ㸪1+4Ȧst2 (2 V /țm)2 ࡢᙧᘧ࡛᰿ྕ࡟ྵࡲࢀ࡚࠸ࡿ㸬ࡑࡢࡓࡵ㸪4Ȧst2 (2V/țm) 2 ࡀᅛ᭷᣺ືᩘࡸῶ⾶ẚ࡟ཬࡰࡍᙳ㡪ࡣ㸪4Ȧst2 (2V/țm) 2 ࡜ 1 ࡜ࡢ኱ᑠ㛵ಀ࡟ࡼࡗ࡚ኚ໬ࡍࡿࡇ࡜ࡀண᝿ࡉࢀࡿ㸬ࡑࡇ࡛㸪4Ȧst2(2V/țm) 2=1 ࡜࡞ࡿ V ࢆ V0 ࡜グࡋ㸪ḟ⠇௨㝆࡛ࡣ Ȧst2 (2V/țm) 2 ࡢᛶ㉁ࢆḟࡢ 2 ࡘࡢሙྜ࡟ศࡅࡿ㸬1) VӐV 0 ࡢሙྜ㸬2) VӑV 0 ࡢሙྜ㸬 9Ӑ9 ࡢሙྜࡢⓎᩓࡢపῶ⟇VӐV 0 ࡢሙྜ㸪1ӑ4Ȧst2 (2V/țm)2 ࡔ࠿ࡽ㸪4Ȧst2 (2V/țm)2ࢆᚤᑠ㔞࡜ࡋ࡚㸪ᘧ(13)ࢆ 1 ḟࡢ࣐ࢡ࣮ࣟࣜࣥᒎ㛤ࢆ⾜࠺࡜㸪ḟᘧࢆᚓࡿ㸬s 2  2] 1LowZ st s  Z st

20          (19)ࡇࡇ࡛ࡇࡇ࡛
] 1Z n1ª2 º«1  1  1 1  4Z 2 §¨ 2V ·¸ » N m        (15)st¨ N ¸ » 2V
«2 2
© m¹ »«¬¼22Z n1§ 2V ·§ 2V · N m (16)¸¸  1  4Z st 2 ¨¨¸¸1  2 1  1  4Z st ¨¨N© m¹© N m ¹ 2V] 2Z n 2ª2 º«1  1  1 1  4Z 2 §¨ 2V ·¸ » N m       (17)st ¨¸ »«
2 2
© N m ¹ » 2V«¬
¼2] 1LowZ st

Z stV2             (20)Nm࡛࠶ࡿ㸬Ⓨᩓࡢ⛬ᗘࢆ㍍ῶࡍࡿࡓࡵ࡟ࡣ㸪᣺ືࡢໟ⤡⥺ࡢⓎᩓࡢ⛬ᗘࢆ⾲ࡍ ȗ1LowȦst ࡀ㸦ṇഃ࡟㸧኱ࡁ࠸᪉ࡀⰋ࠸
࠿ࡽ㸪țm ࡀ኱ࡁ࠸࡯࡝㸪ᘧ(5)࡛ᐃ⩏ࡉࢀࡿ Ȧst ࡀᑠࡉ࠸࡯࡝Ⓨᩓࡢ⛬ᗘࡀ㍍ῶࡉࢀࡿ㸬 9ӑ9 ࡢሙྜࡢⓎᩓࡢపῶ⟇ᘧ(13)ࢆ Vĺࡢᴟ㝈ࢆྲྀࡿ࡜s 2  2] 1HighZ1High s  Z1High20      (21)2§ 2V · N m (18)2 § 2V ·¸¸  1  4Zst 2 ¨¨¸¸1  2 1  1  4Zst ¨¨N© m¹© N m ¹ 2V࡜࡞ࡿ㸬ࡇࡇ࡛

࡛࠶ࡿ㸬ࡇࢀࡽࡀ㸪୍⯡ⓗ࡞㏿ᗘ࡟࠾ࡅࡿᅛ᭷᣺ືᩘ࡜
ῶ⾶ẚ࡛࠶ࡿ㸬] 1HighZ1HighZn 22㸬⪃ᐹ Ᏻᐃᛶ ᘧ(15)ྑ㎶ࡢእഃࡢ᰿ྕෆࡣ 1 ࡼࡾࡶ኱ࡁ࠸㸬ࡑࡢࡓࡵ㸪ȗ1Ȧ1 ࡣᖖ࡟㈇࡛࠶ࡿ㸬ࡋࡓࡀࡗ࡚㸪ᘧ(13)ࡢᴟ࡟ࡼࡗ
࡚㸪࡞ࢇࡽ࠿ࡢእ஘ࡸึᮇ᮲௳ࡀ 0 ࡛࡞࠸ሙྜ㸪 y ࡸ ȥࡣ᣺ືⓗ࡟Ⓨᩓࡍࡿ㸬୍᪉㸪ᘧ(17)ࡣᖖ࡟ṇ࡛࠶ࡿࡓࡵ㸪
ᘧ(14)ࡢᴟ࡟ࡼࡗ࡚ῶ⾶ࡍࡿ㸬ࡑࡇ࡛㸪⺬⾜ື࡟ࡼࡿ୙Ᏻ
－ 75 －2

Z1High4

JN m
14 42r0b

JN m 2r0b

ZV inf(22)(23)

࡛࠶ࡿ㸬᣺ືⓗⓎᩓ᫬ࡢ᫬⣔ิἼᙧࡢໟ⤡⥺ࡢⓎᩓࡢ⛬ᗘࢆ⾲ࡍ ȗ1HighȦ1High ࡟ὀ┠ࡋ㸪ᘧ(22)ࢆ Ȧst ࢆ⏝࠸࡚ኚᙧࡍࡿ࡜] 1HighZ1HighN
1Z st mV2(24)࡜࡞ࡿ㸬ࡋࡓࡀࡗ࡚ VӑV0 ࡢሙྜ㸪Ⓨᩓࡢ⛬ᗘࢆᢚ࠼ࡿࡓࡵࡢᑐ⟇ࡣ㸪țm ࢆࡼࡾᑠࡉࡃ㸪Ȧst ࢆࡼࡾᑠࡉࡃࡍࡿࡇ࡜ࡀ᭷ຠ࡛࠶ࡿ࡜⪃࠼ࡽࢀࡿ㸬㸬ࡲ࡜ࡵᮏㄽᩥࡣ㸪㍯㍈ࡢ⺬⾜ືࢆᐃᘧ໬ࡋ㸪⺬⾜ືࡢⓎᩓ⌧㇟ࢆグྕゎ࡛⾲ࡋࡓ㸬ࡘࡂࡢࡇࡢᘧࢆ⪃ᐹࡍࡿࡇ࡜࡟ࡼ
ࡾ㸪Ⓨᩓࡢపῶἲ࡜ࡋ࡚㸪࠶ࡿ㌴㏿ࡼࡾࡶప㏿ഃ࡛ࡣࢡ࣮ࣜࣉಀᩘ㍯㍈㉁㔞ࢆ኱ࡁࡃ㸪㧗㏿ഃ࡛ࡣᑠࡉࡃࡍࡿࡇ
࡜ࡀ᭷ຠ࡛࠶ࡿࡇ࡜ࢆᣦ᦬ࡋࡓ㸬ㅰ㎡ᮏ◊✲࡟࠶ࡓࡾ㸪Ⲉᇛ኱Ꮫࡢ㐨㎷෸ᩍᤵࡢࡈᣦᑟࢆ㡬ࡁࡲࡋࡓ㸬࠾♩ࢆ⏦ࡋୖࡆࡲࡍ㸬ཧ⪃ᩥ⊩(1) ᪥ᮏᶵᲔᏛ఍⦅,͆㌴୧ࢩࢫࢸ࣒ࡢࢲ࢖ࢼ࣑ࢡࢫ࡜
ไᚚ͇, (1999-2), pp.116-127㸬
(2) 㓇஭ⱥᶞ,͆⮬ື㌴ࡢᖹ㠃㐠ື࡟࠾ࡅࡿ࣮ࣚゅ㏿ᗘ㐍ࡳ᫬ᐃᩘ࡟ࡘ࠸࡚ࡢຊᏛⓗ⪃ᐹ͇, ᪥ᮏᶵᲔᏛ఍
ㄽᩥ㞟(C ⦅)㸪79 ᕳ 801 ྕ (2013-5), pp.1681-1692.
(3) 㓇஭ⱥᶞ,͆ࣇ࢛࣮ࢫࢥࣥࢺ࣮ࣟࣝ࡟࠾࠸࡚୙Ᏻᐃ
㡿ᇦࢆ᭷ࡍࡿ㌴୧ࡢືⓗᣲື࡟ࡘ࠸࡚ࡢᇶ♏ⓗ◊✲͇㸪᪥ᮏᶵᲔᏛ఍ㄽᩥ㞟(C ⦅)㸪81 ᕳ 823 ྕ(2015-3), pp.14-00384.
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