Study on No-undercutting and Its Solution Spaces of Cam Mechanisms with Negative Radius Roller Follower

doi:10.3969/j.issn.1004-132X.2026.03.010

Abstract

Abstract:

The issue of “undercutting” in mechanisms with negative radius follower was highly prone to occur and was very prominent. By discovering the offset curve with opposite direction relationship between the cam pitch curve and cam profile， a new criterion was proposed for no-undercutting based on cam pitch curve. Based on this new criterion， the issue of undercutting was conducted in-depth research. Firstly， a coordinate system was established based on the base circle radius and eccentric offset， and the discrete-traversal method was employed. The inevitable /possible undercutting regions were discovered. The new concepts of non-roller-type/roller-type undercutting were proposed. And the impact of stroke， rise/return phase motion angle， and motion law on inevitable undercutting regions were discussed. Secondly， a vertical numerical axis was established based on roller radius to analyze the distribution characteristics of two types of roller radius values. Then， based on the three-dimensional coordinate system established with the base circle radius， eccentric offset， and roller radius， the solution space and two types of non-solution spaces were obtained for ensuring no-undercutting. Two types of cam profile morphological characteristics， specifically the “swallowtail-shaped- type Ⅰ/type Ⅱ” corresponding to the two types of non-solution spaces were analyzed. This solution space expresses the range of values for the base circle radius， eccentric offset， and roller radius that satisfied the no-undercutting conditions， as well as the minimum size of the cam profile， etc. Finally， the correctness and effectiveness of the proposed theory and method were verified through design cases.

Key words: mechanism with negative radius roller, offset curve with opposite direction, inevitable /possible undercutting region, solution space of no-undercutting

摘要：

负半径滚子机构运动失真问题极易发生、非常突出。通过揭示凸轮理论轮廓、工作轮廓为异向等距曲线，获得了基于凸轮理论轮廓的运动保真新判据。以基圆半径和偏距构造二维坐标系，采取离散-遍历方法，发现必失真/或失真取值域，提出非滚子型/滚子型失真的新概念。讨论了行程、往程/返程运动角以及运动规律对必失真区域的影响。以滚子半径建立垂直数轴，分析了两类滚子半径取值分布特征。基于上述坐标系和数轴构建三维坐标系，获得了运动保真的解域和非解域，深入剖析了两类非解域内的“燕尾交叉Ⅰ型/Ⅱ型”两类凸轮轮廓形貌特征。运动保真解域直观阐释满足运动保真条件的凸轮基本参数取值域、机构最小尺寸解等。通过设计案例验证了所提出理论与方法的正确性和有效性。

关键词: 负半径滚子机构, 异向等距曲线, 必失真/或失真取值域, 运动保真解域

CLC Number:

TH112

HU Zhichao, CHANG Yong, YANG Fufu, WEN Shengxing. Study on No-undercutting and Its Solution Spaces of Cam Mechanisms with Negative Radius Roller Follower[J]. China Mechanical Engineering, 2026, 37(3): 604-611.

胡志超, 常勇, 杨富富, 温圣兴. 负半径滚子直动推杆盘形凸轮机构运动保真及其解域研究[J]. 中国机械工程, 2026, 37(3): 604-611.

Figures/Tables 13

Fig.1 The cam mechanism with negative radius follower

Tab.1 Main motion information of the mechanism

h

/mm

Φ g

Φ r

Φ n

Φ f

s g (φ)

s r (φ)

ω 1

3-4-5

多项式

3-4-5

多项式

顺时针

Tab.1 Main motion information of the mechanism

h

/mm

Φ g

Φ r

Φ n

Φ f

s g (φ)

s r (φ)

ω 1

150°

160°

20°

30°

3-4-5

多项式

3-4-5

多项式

顺时针

Fig.2 The intrinsic relationship between the cam pitch curve and cam profile

Fig.3 Two types of regions of (r0,e)

Fig.4 Comparison of the inevitable undercutting regions DⅠg for different motion information

Fig.5 Distribution characteristics of R values under DⅡ area

Fig.6 Three-dimensional spatial model of (r0,e,R)

Fig.7 The solution space Ω* and two types of non-solution space ΩⅠu、ΩⅡu for ensuring no-undercutting

Fig.8 The cam profile and curvature radius curves under the non⁃solution space of ΩⅠu

Fig.9 The cam profile and curvature radius curves under the non⁃solution of ΩⅡu

Fig.10 The region of undercutting and no-undercutting of e=15 mm

Tab.2 The results for different e value

e

$r 0$ /

$R m i n$ /

Tab.2 The results for different e value

e

/mm

-15

-10

-5

$r 0$ /

144.78

139.36

139.09

144.72

150.08

154.32

156.85

$R m i n$ /

176.18

177.88

179.40

185.89

191.28

195.87

199.84

Fig.11 The physical prototype

References 20

[1]	林梦杰，李延平，常勇. 面向避障和紧凑的浮动滚子推杆盘形凸轮机构尺寸综合问题［J］. 中国机械工程， 2018， 29（20）： 2467-2473.
	LIN Mengjie， LI Yanping， CHANG Yong. Size Synthesis of Disc Cam Mechanisms with Floating Roller Pushrods Aiming at Obstacle Avoidance and Compact［J］. China Mechanical Engineering， 2018， 29（20）： 2467-2473.
[2]	范武，林梦杰，李延平，等. 浮动滚子推杆等跨凸轮机构及其尺寸综合的正-逆映射求交方法［J］. 中国机械工程， 2018， 29（3）： 286-294.
	FAN Wu， LIN Mengjie， LI Yanping， et al. Dimensional Synthesis of Equal-span Cam Mechanisms with Floating Roller Pushrods Based on Technology of Forward-inverse Mapping Intersection［J］. China Mechanical Engineering， 2018， 29（3）： 286-294.
[3]	CARRA S， GARZIERA R， PELLEGRINI M， et al. Criteri per la Progettazione di Camme Speciali （Design Criteria for Special Cams）［C］∥Proceedings of ⅩⅩⅩⅪ AIAS Conference.Parma， 2002.
[4]	CARRA S， GARZIERA R， PELLEGRINI M. Synthesis of Cams with Negative Radius Follower and Evaluation of the Pressure Angle［J］. Mechanism and Machine Theory， 2004， 39（10）： 1017-1032.
[5]	WU Li， LIU C H， CHEN T W. Disc Cam Mechanisms with Concave-faced Followers［J］. Proceedings of the Institution of Mechanical Engineers， Part C： Journal of Mechanical Engineering Science， 2009， 223： 1443-1448.
[6]	贾延龄，王栋梁，张尚盈，等. 一种新型凸轮机构——凹圆弧底从动件盘形凸轮机构的设计（一）［J］. 机械设计， 1998， 15（11）： 39-41.
	JIA Yanling， WANG Dongliang， ZHANG Shangying， et al. Design for a New Type of Cam Mechanisms—Disk Cam Mechanism with Concave Arc Follower（Part 1）［J］. Journal of Machine Design， 1998， 15（11）： 39-41.
[7]	贾延龄，王栋梁，张尚盈，等.一种新型凸轮机构——凹圆弧底从动件盘形凸轮机构的设计（二）［J］.机械设计，1998，15（12）：29-30.
	JIA Yanling， WANG Dongliang， ZHANG Shangying， et al. Design for a New Type of Cam Mechanisms—Disk Cam Mechanism with Concave Arc Follower（Part 2）［J］. Journal of Machine Design， 1998，15（12）：29-30.
[8]	李延平. 正置式凹圆弧底直动从动件盘形凸轮机构的最小尺寸设计［J］. 机械设计， 2002， 19（8）： 30-33.
	LI Yanping. Minimum Dimension Design of Disc Cam Mechanism with Upright Installed Concave Arc Bottomed Translating Follower［J］. Journal of Machine Design， 2002， 19（8）： 30-33.
[9]	LIN Rongfu， CHANG Yong. Mechanical Analysis of Disc Cam Mechanisms with Negative Radius Roller Translating Follower［J］. Applied Mechanics and Materials， 2012， 184/185： 301-306.
[10]	LIN Rongfu， CHANG Yong. Optimization Design of Disc Cam Mechanisms with an Offset Translating Negative Radius Roller Follower［J］. Applied Mechanics and Materials， 2012， 184/185： 774-779.
[11]	QU Chenglong， CHANG Yong. Design of the Disc Cam Profile with Negative Radius Translating Roller Follower［J］. Applied Mechanics and Materials， 2013， 470： 412-415.
[12]	HIDALGO-MARTÍNEZ M， SANMIGUEL-ROJAS E， BURGOS M A. Design of Cams with Negative Radius Follower Using Bézier Curves［J］. Mechanism and Machine Theory， 2014， 82： 87-96.
[13]	SANMIGUEL-ROJAS E， HIDALGO-MARTÍNEZ M. Cam Mechanisms Based on a Double Roller Translating Follower of Negative Radius［J］. Mechanism and Machine Theory， 2016， 95： 93-101.
[14]	HSIEH J F， HSIEH W C. Design of Cam Mechanism with Negative Radius Roller-follower［J］. MATEC Web of Conferences， 2016， 71： 04003.
[15]	HSIEH J F. Design and Analysis of Cam Mechanism with Negative Radius Roller-follower［J］. Transactions of the Canadian Society for Mechanical Engineering， 2016， 40（1）： 89-100.
[16]	吴秋平，常勇，李延平. 负半径滚子直动推杆盘形凸轮机构的运动保真综合［J］. 机械设计与研究， 2016， 32（2）： 26-30.
	WU Qiuping， CHANG Yong， LI Yanping.The Dimension Synthesis of Cam Mechanisms with Negative Radius Follower Based on Movement Fidelity［J］. Machine Design & Research， 2016， 32（2）： 26-30.
[17]	CHANG Yong， WU Qiuping， LI Yanping. Size Synthesis Method for Disc Cam Mechanisms with a Negative Radius Translating Roller Follower［J］. Proceedings of the Institution of Mechanical Engineers， Part C： Journal of Mechanical Engineering Science， 2017， 231（19）： 3624-3634.
[18]	常勇，吴秋平，陈续扬，等. 负半径滚子直动推杆盘形凸轮机构的可视映射尺度综合与比较优势［J］. 机械工程学报， 2017， 53（1）： 36-46.
	CHANG Yong， WU Qiuping， CHEN Xuyang， et al. Dimension Synthesis of Cam Mechanisms with Negative Radius Follower Based on Technology of the Visualization-oriented Mapping and Comparative Advantage［J］. Journal of Mechanical Engineering， 2017， 53（1）： 36-46.
[19]	YAN Zhoudong， CHEN Xinbo， YAN Min， et al. Design and Optimization of a Novel Electronic Mechanical Brake Actuator Based on Cam［J］. Actuators， 2023， 12（8）： 329.
[20]	YAN Zhoudong， CHEN Xinbo， WANG Wei. A Novel Underactuated Cam Mechanism［C］∥Advances in Mechanism and Machine Science. Cham： Springer， 2024： 826-833.