日本機械学会 2012 年度年次大会 [2012.9.9-12] CopyrightⒸ2012 一般社団法人 日本機械学会 [No.12-1] 日本機械学会 2012 年度年次大会講演論文集 〔2012.9.9-12,金沢〕 S111016 平歯車の歯のかみあい剛性に及ぼす歯車加工精度及び歯面修整の影響 * 李 樹庭 *1 Effects of Machining Errors and Tooth Modifications on Mesh Stiffness of a Pair of Spur Gears Shuting LI *1 *1 Shimane Univ. Dept. of Mechanical, Electrical and Electronic Engineering 1060 Nishikawatsu-Cho, Matsue, 690-8504 Japan Three-dimensional, finite element method and a face-contact model of the gear teeth are combined with a mathematical programming method to do loaded tooth contact analysis and mesh stiffness calculations of a pair of spur gears with machining errors and tooth modifications. Effects of the tooth profile deviations, tooth pitch errors, tooth profile modification, tooth lead crowning and relieving on the mesh stiffness are investigated. It is found that the tooth profile deviations, tooth pitch errors and tooth profile modifications have significant effects on the mesh stiffness. It is also found that the lead crowning and relieving make the mesh stiffness smaller, but have no effects on load sharing ratios of the gears. Since the lead crowning makes the maximum contact stress of the teeth much greater and the lead relieving make the teeth have edge loads, it is necessary to pay a special attention to the quantity of the lead crowning and the length of the lead relieving, not to make the quantity of the crowning and the length of the relieving too larger. Key Words : Gear, Spur Gear, Mesh Stiffness, Contact Analysis, Machining Errors, Tooth Modification 1. 緒 言 歯車の歯のかみあい剛性は歯車の振動特性を決める重要なパラメータの一つである.しかし,歯車の歯のかみ あい剛性に影響を及ぼす要因が多く,例えば,歯車の加工精度,組立精度,歯面修整,ベアリングの支持剛性, ハウジングの支持剛性と歯車軸のたわみなどの影響を受けるので,歯車の歯のかみあい剛性に関する理論研究及 び実験研究が難しく,この問題はまだ完全に解決されていないのが現状である. 誤差を持つ歯車の歯のかみあい剛性に関する研究は久保ら (1) と梅沢・北條ら (2) の研究がある.李 (3-5) は三次 元有限要素法,線形計画法と歯の面接触モデルを併用することにより組立誤差を持つ一対の薄肉平歯車の作用線 方向の総変形を求める方法を提案した.そして歯車の作用線上の総荷重をこの総変形で除することにより,組立 誤差,加工誤差と歯面修整を持つ一対の平歯車の歯のかみあい剛性の計算法を提案した (4) .本研究では,これら の研究を更に発展させて,一対の平歯車の歯のかみあい剛性に及ぼす歯車の歯形誤差,ピッチ誤差,歯形修整, 歯筋クランニングとレーリビングの影響を理論上更に詳しく検討し,その結果を次に述べる. 2. 歯車の加工精度及び歯面修整について 2・1 歯車の加工精度について 本研究において、歯車の加工精度を歯面の三次元歯形誤差と歯のピッチ誤差に分けて考慮するようにしている. ホブ切り歯車の三次元歯形誤差の形状は工作機精度と加工条件によって様々であるが,図1(a)と(b)に示す 二つのパターンをホブギリ小歯車と大歯車の三次元歯形誤差として一対の平歯車の歯のかみあい剛性解析に用い た.図 1 において,横軸と奥行き軸はそれぞれ歯の歯筋方向と歯たけ方向の無次元寸法であり,縦軸は歯面の三 次元歯形誤差である.ピッチ誤差については小歯車に法線ピッチ誤差を与えるようにしている. *1 正員,島根大学(〒690-8504 松江市西川津町 1060) E-mail: [email protected]
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Effects of Machining Errors and Tooth Modifications on Mesh Stiffness of a Pair of Spur Gears
Shuting LI*1
*1 Shimane Univ. Dept. of Mechanical, Electrical and Electronic Engineering 1060 Nishikawatsu-Cho, Matsue, 690-8504 Japan
Three-dimensional, finite element method and a face-contact model of the gear teeth are combined with a mathematical programming method to do loaded tooth contact analysis and mesh stiffness calculations of a pair of spur gears with machining errors and tooth modifications. Effects of the tooth profile deviations, tooth pitch errors, tooth profile modification, tooth lead crowning and relieving on the mesh stiffness are investigated. It is found that the tooth profile deviations, tooth pitch errors and tooth profile modifications have significant effects on the mesh stiffness. It is also found that the lead crowning and relieving make the mesh stiffness smaller, but have no effects on load sharing ratios of the gears. Since the lead crowning makes the maximum contact stress of the teeth much greater and the lead relieving make the teeth have edge loads, it is necessary to pay a special attention to the quantity of the lead crowning and the length of the lead relieving, not to make the quantity of the crowning and the length of the relieving too larger.
(a) Deviations on pinion (b) Devaitions on gear (a) Profile modification (b) Lead crowning (c) Lead relieving Fig.1 Tooth profile deviations of hob-cutting gears Fig.2 Methods used for tooth modifications
Wid
th
Plane of action
Geometric contact line
Reference faceReference line
Reference point
k
k k'
Wid
th
k
0
Y0
0
X0
Geometric contact point
Line of action
Parallel Line
Pair of assumedcontact points
P
P
(a) Contact tooth surface of Gear 1 (b) Contact teeth of Gear 1 and Gear 2
Fig.3 Face-contact model of a pair of spur gears used for mesh stiffness analysis
(a) Pinion FEM mesh (b) 3D view of the gears (c) Two pair tooth contact (d) One pair tooth contact Fig.4 Three dimensional, view and the section view of FEM mesh dividing patterns of the gears
0 5 10 15 20 25 30 35 40
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08Geometric contact line
Ideal gears & Tooth 1
Tooth longitudinal dimension mm
Con
tact
wid
th m
m
393.8 -- 450.0
337.5 -- 393.8 281.3 -- 337.5
225.0 -- 281.3
168.8 -- 225.0
112.5 -- 168.8
56.25 -- 112.5 0 -- 56.25
Fig. 5 Comparison of the mesh stiffness Fig. 6 Contact stresses of the gears without errors and modifications
(a) Only pinion having deviations (b) Only gear having deviations (c) Gear & pinion having deviations Fig.7 Contact stress distributions of the gears with tooth profile deviations
(a) The pinion with minus pitch errors (b) The pinion with plus pitch errors
Fig. 8 Mesh stiffness of the gears with pitch errors
0 5 10 15 20 25 30 35 40
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08 Geometric contact line
Tooth 1Pitch error=-2μm
Tooth longitudinal dimension mm
Con
tact
wid
th m
m
481.3 -- 550.0 412.5 -- 481.3
343.8 -- 412.5 275.0 -- 343.8
206.3 -- 275.0 137.5 -- 206.3
68.75 -- 137.5 0 -- 68.75
0 5 10 15 20 25 30 35 40
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08Geometric contact line
Tooth 1Pitch error=+2μm
Tooth longitudinal dimension mm
Con
tact
wid
th m
m
245.0 -- 280.0 210.0 -- 245.0
175.0 -- 210.0 140.0 -- 175.0
105.0 -- 140.0 70.00 -- 105.0
35.00 -- 70.00 0 -- 35.00
(a) The pinion with minus pitch errors (b) The pinion with plus pitch errors Fig. 9 Contact stress distributions of the gears with pitch errors
0 5 10 15 20 25 30 35 40
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
Geometric contact line
Tooth 1Tooth profile modification=6μm
Tooth longitudinal dimension mm
Con
tact
wid
th m
m
437.5 -- 500.0
375.0 -- 437.5 312.5 -- 375.0
250.0 -- 312.5
187.5 -- 250.0 125.0 -- 187.5
62.50 -- 125.0
0 -- 62.50
(a) Mesh stiffness curves (b) Contact stress distribution Fig. 10 Mesh stiffness and tooth contact stresses of the gears with profile-modified pinion
(a) Lead crowned pinion (b) Lead relieved pinion Fig. 11 Mesh stiffness curves of the gears with crowned pinion and relieved pinion