文章编号 :1004 - 2539 (2004) 05 - 0026 - 03
直齿圆柱齿轮变位系数的优化选择
(安徽建筑工业学院数理系 , 合肥 230022) 梁华琪
摘要 通过对影响圆柱齿轮变位系数的接触强度原则、弯曲强度原则、抗胶合能力和耐磨性能等因
素的分析 ,提出齿轮变位系数优化选择的数学模型及其求解方法。
关键词 圆柱齿轮 变位系数 模型 优化
引言
用齿条型刀具滚切齿轮时 ,为了避免根切或齿厚
太小 ,刀具的基准线和轮坯的分度圆之间往往偏离一
段距离 (即变位量) ,用 xm
表
关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf
示 , x 为变位系数。当刀
具离开轮坯中心时 , x 取正值 ;反之 , x 取负值。变位
系数的大小对齿轮齿根强度、齿面接触强度、抗胶合强
度和传动噪声有复杂的影响。对于某种强度因素最适
用的变位系数 ,对于其他强度因素就不一定适用。所
以 ,适用于任何尺寸和材料的齿轮变位系数或适用于
任何使用条件的变位系数是不存在的。只能根据给定
的模数、齿数、材料和使用条件 ,考虑该齿轮最可能产
生的失效形式 ,选择适当的变位系数。
变位系数的选择有很多方法。常用的有列表法、
封闭图法和优化选择法。本文主要讨论在满足接触强
度和弯曲强度条件下 ,同时考虑齿轮抗胶合能力和耐
磨性能 ,如何优化选择变位系数。
1 变位系数的选择原则及其目标函数
的建立[1 ]
在齿轮传动中 ,选择变位系数的一般原则如下
(a) 为了提高齿轮接触强度 ,防止齿面疲劳点蚀或
剥伤 ,变位系数的选择应使啮合点处的齿廓综合曲率
半径增大。为此 ,应选用尽可能大的 xΣ ,其目标函数
为
maxf 1 ( x) = xΣ = x1 + x2 (1)
两轮的变位系数之和 x ∑可由无侧隙啮合方程式
导出 ,即
x1 + x2 =
z1 + z2
2tanα (invα′- invα) (2)
式中 α ———齿轮分度圆上的压力角
α′———两齿轮的啮合角
z1、z2 ———两齿轮的齿数
当齿轮传动的实际中心距 a′由结构给定时 ,啮合
角为
α′= arccos ( m ( z1 + z2) cosα2α′ ) (3)
(b) 为了提高弯曲强度 ,减小齿根应力集中 ,应尽
量减小齿形系数并使两齿轮的弯曲强度趋于平衡 ,其
目标函数为
min f 2 ( x) =σF1 - σF2 (4)
式中[2 ]
σF1 =
Ft
bm KA KV KFβKFαYFa1 YSa1 Yε
σF2 =
Ft
bm KA KV KFβKFαYFa2 YSa2 Yε
(5)
式中 KA ———使用系数
KV ———动载系数
KFβ ———齿向载荷分布系数
KFα ———齿间载荷分配系数
YFai ———齿形系数 ( i = 1 ,2)
YFai = f ( x i) ( i = 1 ,2)
YSai ———应力修正系数 ( i = 1 ,2)
YSai = f ( xi) ( i = 1 ,2)
Yε ———弯曲强度计算的重合度系数
Yε= 0. 25 + 0. 75/εα= f ( x1 + x2)
εα ———重合度
其中 , KA 、KV 、KFβ、KFα的值可查参考文献[3 ]。
(c) 为了提高齿轮抗胶合能力和耐磨性能 ,应使啮
合齿在开始啮合时主动齿轮齿根处的滑动系数η1 与
啮合终了时从动齿轮齿根处的滑动系数η2 相等 ,其目
标函数为
min f 3 ( x) =η1 - η2 (6)
因滑动系数是滑动弧与齿廓所走过的弧长之比 ,
62 机械传动 2004 年
© 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
啮合始点是主动轮的齿根和从动轮的齿顶相接触的
点 ,啮合终点是主动轮的齿顶和从动轮的齿根相接触
的点 ,经推导可得[3 ]
η1 =
tanαa1 - tanα′
(1 + z1
z2
) tanα′- tanαa1
(1 + z1
z2
)
η2 =
tanαa2 - tanα′
(1 + z2
z1
) tanα′- tanαa2
(1 + z2
z1
)
(7)
式中的αa1和αa2分别为主动轮和从动轮齿顶圆上的压
力角 ,按下式计算
αai = arctan ( rbi/ rai)
rbi = 0. 5 mzicosα
rai = m (0. 5 zi + ha 3 + x i - σ)
σ= x + x - y
y = ( a′- a) / m
( i = 1 ,2) (8)
式中 rbi ———两轮基圆半径
rai ———两轮齿顶圆半径
h 3a ———齿顶高系数
σ ———齿顶高变动系数
y ———分度圆变动系数
由此可见 ,两齿轮齿根的滑动系数η1、η2 和两轮
的弯曲应力σF1、σF2与两轮的变位系数有关。在实际
中心距 a′给定的情况下 , x1 和 x2 中仅有一个是独立
的。若取 x1 为设计变量 ,η1、η2、σF1、σF2均为 x1 的函
数。那么 x1 优化的总目标函数为
min f ( x1) = - w1 ( x1 + x2) | w2 (σF1 -
σF2) | w3 (η1 - η2) (9)
式中 , wi ( i = 1 ,2 ,3) 为加权因子 ,它随各个分目标函数
在相对重要程度的差异而变化。
2 选择变位系数的限制条件
(1) 齿根根切对变位系数的限制
用齿条形刀具加工MATCH_
word
word文档格式规范word作业纸小票打印word模板word简历模板免费word简历
_1713963441403_0齿轮时 ,切制齿数 z < zmin
的标准齿轮将发生根切 ,其中
zmin = 2 h 3a / sin2α
另外 ,当切制变位量不够大的正变位齿轮和变位
量过大的负变位齿轮时也会发生根切。这种不使变位
齿轮产生根切的变位系数的最小值称为最小变位系
数 ,以 xmin表示 ,应使
x ≥xmin = ha 3 ( zmin - z) / zmin = ha 3 - 0. 5 zsin2α
(10)
(2) 齿轮齿顶变尖对变位系数的限制
随着变位系数 x 的增大 ,齿形会逐渐变尖。为了
保证齿顶的强度 ,
要求
对教师党员的评价套管和固井爆破片与爆破装置仓库管理基本要求三甲医院都需要复审吗
齿顶厚 S a ≥(0. 25 - 0. 4) m ,齿
轮材料组织均匀者取下限 ,齿面经硬化处理者取上限。
如果不满足这一条件时 ,则应适当地减小变位系数 ,重
新进行设计。那么齿顶厚应为
S a = S
ra
r
- 2 ra (invαa - invα) (11)
式中
r = 0. 5 mz
S =πm/ 2 + 2 xmtanα
(3) 重合度对变位系数的限制
齿轮的重合度ε随着变位系数 x 的增大而减小。
选择变位系数时 ,应保证齿轮传动的重合度大于等于
许用重合度[ε]。设εa 为端面重合度 ,一般应使ε=
εa ≥1. 2。εa 的计算式为
εa = 12π[ z1 (tanαa1 - tanα′) + z2 (tanαa2 - tanα′) ] ≥1. 2
(12)
(4) 齿轮干涉对变位系数的影响
一对齿轮啮合传动时 ,如果一个齿轮的齿顶的渐
开线与另一齿轮齿根的过渡曲线接触 ,由于过渡曲线
不是渐开线 ,故两齿廓在接触点的公法线不能通过固
定的节点 ,因而引起传动比的变化 ,还可能使两轮卡住
不动 ,这种现象称为“过渡曲线干涉”。当变位系数 x
的绝对值过大时 ,会产生干涉。选择变位系数时 ,要保
证任一齿轮的齿顶不与相啮合齿轮的齿根产生过渡曲
线干涉。当用齿条形刀具加工标准齿轮时 ,外啮合直
齿圆柱齿轮 1 不产生干涉的条件为
tanα′- z2
z1
(tanαa2 - tanα′) ≥tanα-
4 ( ha 3 - x1)
z1sin2α (13)
齿轮 2 不产生干涉的条件为
tanα′- z1
z2
(tanαa1 - tanα′) ≥tanα-
4 ( ha 3 - x2)
z2sin2α (14)
3 数学模型的建立及求解方法[4 ]
3. 1 目标函数的转化
由于目标函数 f ( x1) 是由 3 个分目标函数组成 ,
它们在量级和性质上有较大差别。为了求解方便 ,需
将这一多目标规划问题按一定规则转化为单目标规划
问题 ,然后才能求解。转化方法如下
72第 28 卷 第 5 期 直齿圆柱齿轮变位系数的优化选择
© 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
(1) 在边界约束条件下 ,将 3 个分目标函数转化
为 0~1 之间的规格化目标函数[5 ] 。采用正弦函数 y
= x/ 2π- sin x (0 ≤x ≤2π) ,各个分目标函数 f i ( x1) 的
设计指标变动范围为 pi ≤f i ( x1) ≤qi ,式中 pi , qi 分别
为设计指标的上、下限。那么相应于 f i ( x1) 的转化函
数 f ri ( x1) 的自变量 x1 i为 x1 i = ( f i ( x1) - pi) 2π/ ( qi -
pi) ,于是规格化目标函数 f ri ( x1) 为
f ri ( x1) =
x1 i
2π - sin x1 i =
f i ( x1) - pi
qi - pi
- sin ( f i ( x1) - piqi - pi 2π)
( i = 1 ,2 ,3)
(2) 加权因子 w i 选择采用目标函数的容限法求
解 ,由于目标函数变动范围为 pi ≤f i ( x1) ≤qi ( i = 1 ,2 ,
3) ,那么各项目标的容限 Δf i = ( qi - pi ) / 2 ( i = 1 , 2 ,
3) ,加权因子 w i = 1/ (Δf i) 2 ( i = 1 ,2 ,3) 。
(3) 这样统一多目标函数式 (9) 可改写成
min f ( x1) = - w1 f r1 ( x1) + w2 f r2 ( x1) + w3 f r3 ( x1) (15)
3. 2 约束条件的确定
根据对选择变位系数的限制条件的分析 ,目标函
数受约于
g1 ( x1) = x1 ≥xmin = ha 3 - 0. 5 zsin2α
g2 ( x1) = S a = S
ra
r
- 2 ra (invαa - invα) ≥0. 25
g3 ( x1) =εα= 12π[ z1 (tanαa1 - tanα′) +
z2 (tanαa2 - tanα′) ] ≥1. 2
g4 ( x1) = tanα′- tanα-
z2
z1
(tanαa2 - tanα′) +
4 ( ha 3 - x1)
z1sin2α ≥0
g5 ( x1) = tanα′- tanα-
z1
z2
(tanαa1 - tanα′) +
4 ( ha 3 - x2)
z2sin2α ≥0
3. 3 数学模型的建立
由以上分析可知 ,求解 x1 的数学模型为
min f ( x1)
gu ( x1) ≤0
( u = 1 ,2 , ⋯,5 ;) (16)
3. 4 复合形法的求解
复合形法是求解约束非线性最优化问题的一种重
要的直接方法。它来源于求解无约束非线性最优化问
题的单纯形法 ,实际上是单纯形法在约束问题中的发
展。在用于求解约束问题的复合形法中 ,复合形各顶
点的选择和替换 ,不仅要满足目标函数值的下降 ,还应
当满足所有的约束条件。复合形法的求解过程如下
①输入初始数据 m、z1、z2、ha 3 、a′、α; ②首先在
可行域产生均匀分布的伪随机数 ,确定初始复合形的
第 1 个顶点 ,并检查是否在可行域内。如果不在则重
新产生伪随机数并再选点 ,直到第 1 个顶点在可行域
为止。用同样的方法产生其它 9 个顶点 ,从而得到复
合形的全部顶点 ; ③计算中心点。计算所有各顶点的
函数值 ,找出函数值最大 x1 h 的点并舍去 ,计算其余各
顶点的中心 ,即形心点 x1 c ; ④求出反射点 x1 r = x1 c +
k ( x1 c - x1 h) ; ⑤计算反射点的目标函数值 ,并与最坏
点的目标函数值进行比较。根据比较的结果调整反射
点 ,直到满足所有约束条件为止。重复步骤 ③~ ⑤,直
到复合形中各顶点距离小于预先给定的精度要求为
止。
4 设计示例
某齿轮厂要设计一对直齿圆柱齿轮 , m = 4mm ,α
= 20°, ha 3 = 1 , z1 = 20 , z2 = 80 , 实际中心距 a′=
225. 35mm ,传递功率 P = 20kW。试从接触强度、弯曲
强度和抗胶合能力方面设计此齿轮。
在该例中 ,经计算得到 :2. 23 ≤f 1 ( x1) ≤4. 48 , 0 ≤
f2 ( x1) ≤32 , - 0. 4 ≤f 3 ( x1 ) ≤4 ; 同时也得到 : w1 =
0. 79 , w1 = 0. 004 , w1 = 0. 22 ;经复合形优化得到 : x1 =
0. 879 , x2 = 1. 792。如采用传统设计 ,可得 : x1 = 0. 766 ,
x2 = 1. 464。通过此方法选择变位系数的的优越性 ,
有待在实际使用中加以验证。
参 考 文 献
1 邱宣怀主编. 机械设计. 北京 :高等教育出版社 ,1991 ,167~204
2 成大先主编. 机械设计手册 (第三版 ,第三卷) . 北京 :化学工业出版
社 ,1997
3 孟彩芳编著. 机械原理电算分析与设计. 天津 :天津大学出版社 ,
2000 ,144~159
4 刘惟信主编. 机械最优化设计. 北京 :清华大学出版社 ,1993 ,230~
233
5 陈立周主编. 机械优化设计方法. 北京 :冶金工业出版社 ,1999 ,151
~153
收稿日期 :20031229 收修改稿日期 :20040219
作者简介 :梁华琪 (1970 - ) ,男 ,安徽肥东人 ,实验师
82 机械传动 2004 年
© 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
ABSTRACTS & KEY WORDS
JOURNAL OF MECHANICAL TRANSMISSION Vol. 28. No. 5 , 2004
The Comparative Study of the Bending Stress and Elastic Deforma2
tion Calculation of Gear tooth
Zhou Changjiang , Tang Jinyuan , Wu Yunxin(1)
⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
⋯⋯⋯⋯⋯⋯
Abstract The progress character and application of the calculation
methods of the root stress and elastic deformation for gear tooth are dis2
cussed in detail , such as the analytical method , the experimental
mothed and the numerical methed. Several typical methods are compar2
ied. The FEM is recommended as the core of the CAE of gear transmis2
sion design.
Key words : Root stress Tooth deformation Comparative study
Study on the Profile Equation of the Rigid Gear of the Harmonic
Chain Drive Song Guanying , Chen Shuchang(7)⋯⋯⋯⋯⋯⋯⋯
Abstract The harmonic drive is regarded as a great breakthrough of
the mechanical drive. But the harmonic gear drive has some insur2
mountable defects. This paper introduces a new type of harmonic drive
- - the harmonic chain drive. And for the first time , this paper estab2
lishes the profile equation of the rigid gear on the basic of differential
geometry and mesh theory.
Key words : Harmonic chain drive Mesh theory Profile equation
Fault Diagnosis of Rolling Bearing Based on AR model and Radial
Basis Function Neural Networks
Lu Shuang , Hou Yueqian , Tian Ye(10)
⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
⋯⋯⋯⋯⋯⋯⋯⋯⋯
Abstract Radial basis function neural network is a type of three - lay2
er feedforward non - linear network. It has many good properties , such
as powerful ability for function approximation , classification. In this pa2
per , in the light of the merit of radial basis function neural network and
on the basis of the feature analysis of vibration signal of rolling bearing ,
AR model is presented by using time series method. Radial basis func2
tion neural networks is established based on AR model parameter. In
the light of the theory of radial basis function neural networks , fault pat2
tern of rolling bearing is recognized correspondingly. Theory and experi2
ment shows that the recognition of fault pattern of rolling bearing based
on AR model and radial basis function neural networks theory is avail2
able and its precision is high.
Key words : Rolling bearing Vibration signal AR model RBF
neural networks Fault diagnosis
The Finite Element Solution of Load Distribution among Teeth in
Meshing Process of Spur Gear
Bao Jiahan , Zhang Yuhua , Xue Jiaguo(14)
⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
⋯⋯⋯⋯⋯⋯⋯⋯
Abstract The condition of contact stress and load distribution among
teeth has been studied which varies with the engaging position in the
meshing process of spur gear. The results indicate that friction force can
enlarge contact stress and load share of the former meshing teeth.
Key words : Spur gear Load distribution among teeth Finite ele2
ment Contact stress
The Fuzzy Reliability Optimum Design Of Cycloid Steel Ball Plane2
tary Transmission An Zijun , Wang Guangxin(17)⋯⋯⋯⋯⋯⋯⋯
Abstract Based on Fuzzy Reliability theory , the mathematic model of
fuzzy reliability optimum design on the end face engagement cycloid
steel ball planetary transmission is established. The fuzzy reliability op2
timum design for the transmission example is solved by SUMT method.
This method is of great theory and practice significance.
Key words : Cycloid Steel ball Planetary transmission Fuzzy re2
liability Optimum design
The Optimization for Mathematic Characteristies of Radius Varia2
tion Epicycloids and Optional Data of Cycloidal Drives
Liu Jibo (20)
⋯⋯⋯⋯
⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
Abstract From the same conditions of circurnscribed epicycloids and
inscribed epicycloids , it can be derived that both the central distance of
epicycloids generating circle & base circle and radius variation ratio are
as the formula of known data. From the basic data of cycloidal drives
(pin drive in shorter form later) , it can be derived that both the curtate
ratio K1 and the number of pin gear z are basic data of known data
when the eccentric distance a = 1. But the cycloidal profile and its for2
mula of basic data under the optional eccentric distance is only the lin2
ear amplification and linear reduction of the multiple based on eccentric
distance a. From the data of gear pin radius and the min , cycloidal
curvature radius , it can be derived that the maximam load capacity of
cycloidal drive is the optimized data of gear pin radius ratio. From the
load eapacity of cycloidal drive and the size of drive device , it can be
derived the optimized method , which can be generated as calculation of
data K1 with the maximun data (ηvmax) of the efficiency and size , of
getting maximam load capacity K1 by the cycloidal drive wit minimam
size.
Key words : Pin radius ratio Curtate ratio Load capacity Load
capacity factor
The Optimization for the Addendum Modification Coefficient of
英文摘要 2004 年
© 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
Spur Gear Liang Huaqi (26)⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
Abstract Based on the evaluation of the influnce of the addendum
modification coefficient on the surface durability , bending strength ,
scuffing resistance and wear resistance of spur gear , the mathematical
model for the optimization of addendum modification coefficient is estab2
lished and its solution proce dure is given.
Key words : Spur gear Addendum modification coefficient Model
Optimization
The Tooth Modification for the Low Speed Open Gear of Heavy
Load Wu Xiue , Zhan Yan(29)⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
Abstract The theoretical calculation of the tooth modification for the
low speed gear of heavy load is given , and the experiments for the
bending strength and wear resistance are carried out. It is shown that
the proper tooth modification is helpful for the performance of this kind
of gear.
Key words : Tooth modification Wear resistance Bending
strength Open gear
Auto - modeling of Gear Based on Pro/ Engineer
Wang Sanwu ,Jing Renkun , Yu Xuyang(35)
⋯⋯⋯⋯⋯⋯
⋯⋯⋯⋯⋯⋯⋯⋯
Abstract Based on pro/ Engineer , a design method of multi - function
parameter involute cylinder gear is introduced. Spur gear , Helical gear
and shifeed gear can be generated by the method. Furthermore , auto -
modeling of gear can be realized with the help of developing tool Pro2
gram of the method.
Key words : Pro/ Engineer Involute Gear Modeling Feature
Parametric Design and Dra wing Based on VBA
Ding Qizhi , Fu Jinghong (38)
⋯⋯⋯⋯⋯⋯⋯
⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
Abstract The principle for the parametric design based on the Object
Oriented language VBA is presented , and the practical method to realize
it and the key technology are discussed with an example.
Key words : VBA Parametric design CAD
The Software Development for the Mechanism Signs Based on the
Icon Menu
Lv Shouxiang , He Wei , Liu Yansong , Wu Chunying(40)
⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
⋯⋯
Abstract A software package is developed to make diagram drawing of
mechanisms under the Auto CAD 2000. The visualized sign , icon
menu , pull - down menu are used to help the designer to make the in2
teractive drawing of mechanism diagrams at high speed.
Key words : Mechanism diangram LISP Icon menu
The CAD Analysis System for the Accuracy of Cylindrical Gear
Jiang Shiping , Li Xijian , Huang Manling , Xia Zhixin(43)⋯⋯
Abstract Frist , theory analysis of gear accuracy is introduced. then
component part and main function of design softwere are analyzed , fi2
nally feasibility of the softwere is verified with an example. Based on
Visual Basic , a design softwore of gear accuracy is developed , using the
software makes the design work easy and convenient. The software is
easy to operate and can be used prompty and correctly to determine the
accuracy class of three tolerance groups , the limit deviation of tooth
thickness , the check items and the tolerance values.
Key words : Gear accauacy Backlash Tolerance group Check
group
One Modern Version Heavy Load Friction Clutch
Liang Yide , Liang Huifang(55)
⋯⋯⋯⋯⋯⋯
⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
Abstract A modern version friction clutch is introduced. Friction co2
efficient of the clutch is 10 - 30 times larger than that of the common
friction clutch. There are a lot of advantage of the clutch , such as the
volume is small , power transmitted is great , the structure is simple ,
technological behaviour is good , cost is lows , life is long. Therefore , It
is an ideal friction clutch of modern version.
Key words : Mordern version Heavy load Friction Clutch
The Stress Distribution of the Rotary Shell parts Under the Axial
Symmetrical Loads
Liang Zhiguo ,Wu Chunying ,Wang Ningxia(59)
⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯
⋯⋯⋯⋯⋯⋯
Abstract On the basis of the stress distribution of the rotary thin shell
parts under the symmetrical axial loads , the characteristics of stress dis2
tribution for the several typical thin shell parts are put forward and the
simple design approach to the shell parts with the uniform shear stresses
is presented.
Key words : Thin shell parts Stress analysis Uniform shear stress2
es
The History and development of Gear Technology of China
Wu Xiaoling , Yuan Lijuan , Xu Wanli(62)
⋯⋯
⋯⋯⋯⋯⋯⋯⋯⋯
Abstract The centuries - old history of gear technology of China is re2
viewed. The gear technology has got a great progress since the founding
of the People’s Republic of China. Finally , the prosperous future of
gear manufacture industry of China is predicted.
Key words : Gear technology History Gear manufacture industry
2
© 1994-2006 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net