Mechanisms ( 機 構 學 ) 原 名 : 機 動 學, 即 機 械 運 動 學 (kinematics) 之 簡 稱 Text: K. J. Waldron and G. L. Kinzel, 2004, Kinematics, Dynamics, and Design of Machinery, 2 nd ed., John Wiley & Sons. ( 歐 亞 ) Reference: 1. 顏 鴻 森, 機 構 學, 東 華 書 局. 2. G. Bögelsack, F. J. Gierse, V. Oravsky, J. M. Prentis, and A. Rossi, 1983, Terminology for the Theory of Machines and Mechanisms, Pergamon Press. C. F. Chang, KUAS ME 1 Contents Basic Concepts: ( 基 本 觀 念 ) Chapter 1 Introduction Linkages: ( 連 桿 組, 連 桿 機 構 ) Chapter 2 Graphical Position, Velocity, and Acceleration Analysis for Mechanisms with Revolute Joints of Fixed Slides Chapter 3 Linkages with Rolling and Sliding Contacts and Joints on Moving Sliders Chapter 4 Instant Centers of Velocity Chapter 5 Analytical Linkage Analysis Cam and Gears: ( 凸 輪 和 齒 輪 ) Chapter 8 Profile Cam Design Chapter 10 Spur Gears Chapter 11 Helical, Bevel, and Worm Gears Chapter 12 Gear Trains We will focus on the so-called Planar Motion Motion of links whose points describe curves located in parallel planes. [ 桿 件 上 各 點 之 運 動 皆 在 同 一 平 面 或 其 平 行 平 面 上 ] C. F. Chang, KUAS ME 2 國 立 高 雄 應 用 科 大 機 械 系 1
Chapter 1. Introduction Definition of Mechanisms Mechanisms are assemblages of rigid member connected together by joints. (p.3) ( 機 構 係 由 機 件 與 接 頭 所 構 成 之 可 動 組 合 ) C. F. Chang, KUAS ME 3 Mechanism Vs Machine Links Constrained motion? mechanism Joints Mechanisms transfer motion to to one oneor ormore output members Machine transfer motion and and useful work worktoto one oneor ormore output members (( 機 器 為 可 輸 出 有 用 之 功 的 機 構 )) Power Controller Output effective work? Constrained motion: motion: (( 各 機 件 皆 產 生 確 切 且 可 預 期 之 運 動 )) machine C. F. Chang, KUAS ME 4 國 立 高 雄 應 用 科 大 機 械 系 2
Terminology For MMT Kinematic chain [ 運 動 鏈 ] Assemblage of links and joints. Mechanism [ 機 構 ] Kinematic chain with one of its components (link or joint) connected to the frame and with definite motion 具 有 確 切 運 動 且 至 少 有 一 桿 固 連 於 機 架 之 運 動 鏈 System of bodies designed to convert motions of and forces on one or several bodies into constrained motions of and forces on other bodies (MMT) 一 支 或 多 支 桿 件 之 運 動 和 受 力 轉 換 為 其 他 桿 件 之 拘 束 運 動 和 受 力 Machine( 機 器 ) Device performing mechanical motion to transform and transfer energy, material and information 是 一 種 執 行 機 械 運 動 的 裝 置, 用 來 變 換 和 傳 遞 能 量, 材 料 與 資 訊 C. F. Chang, KUAS ME 5 Terminology For MMT Link [ 連 桿, 機 件 ] 1. Mechanism element (component) carrying kinematic pairing elements [ 機 構 元 件, 用 來 帶 動 以 運 動 對 連 接 之 元 件 ] 2. Element of a linkage. [ 連 桿 組 之 元 件 ] Joint [ 接 頭 ] The physical embodiment of kinematic pair.[ 運 動 對 之 具 體 化 身 ] Kinematic pair [ 運 動 對 ] Contacting elements of links permitting their constrained relative motion. [ 桿 件 間 之 接 觸 部 份, 它 使 桿 件 之 間 產 生 拘 束 的 相 對 運 動 ] Lower pair Kinematic pair which is formed by surface contact of its elements. [ 經 由 面 接 觸 所 構 成 之 可 動 連 接 ] High pair Kinematic pair which is formed by point or line contact of its elements [ 經 由 點 或 線 接 觸 所 構 成 之 可 動 連 接 ] Connectivity(Degree of freedom of a joint): the number of independent coordinates needed to describe the relative positions of pairing elements [ 確 定 兩 桿 件 之 相 對 位 置 所 需 之 獨 立 參 數 的 數 目 ] C. F. Chang, KUAS ME 6 國 立 高 雄 應 用 科 大 機 械 系 3
Lower Pair Joints [ 六 種 常 見 之 低 對 接 頭 ] C. F. Chang, KUAS ME 7 Name: 1. Revolute Pair (R) [ 旋 轉 對 ] 1. Revolute hinge 2. turning pair Letter symbol: R Connectivity (DOF) : 1 Dof Dofof ofkinematic pair pair (connectivity) = the thenumber of ofindependent coordinates needed neededtoto describe the the relative relativepositions of ofpairing elements 接 頭 之 自 由 度 = 確 定 兩 桿 件 之 相 對 位 置 所 需 之 獨 立 參 數 的 數 目 C. F. Chang, KUAS ME 8 國 立 高 雄 應 用 科 大 機 械 系 4
2. Prismatic Pair (P) [ 滑 行 對 ] Name: 1. Prismatic joint 2. Slider 3. Sliding pair Letter symbol: P Connectivity (DOF): 1 C. F. Chang, KUAS ME 9 3. Helical Pair (H) [ 螺 旋 對 ] Name: 1. Screw joint 2. Helical joint 3. Helical pair Letter symbol: H Connectivity (DOF) : 1 C. F. Chang, KUAS ME 10 國 立 高 雄 應 用 科 大 機 械 系 5
Name: 4. Cylindrical Pair (C) [ 圓 柱 對 ] 1. Cylindrical Joint 2. Cylindrical pair Letter symbol: C Connectivity (DOF) : 2 C. F. Chang, KUAS ME 11 5. Spherical Pair (S) [ 球 面 對 ] Name: 1. Spherical joint 2. Ball joint 3. Spherical pair Letter symbol: S Connectivity (DOF) : 3 C. F. Chang, KUAS ME 12 國 立 高 雄 應 用 科 大 機 械 系 6
6. Planar Pair (R) [ 平 面 對 ] Name: 1. Planar joint 2. Planar pair Letter symbol: R Connectivity (DOF) : 3 C. F. Chang, KUAS ME 13 Replacement of a Lower Pair Joint by a combination of Higher Pair r Joints In order to reduce the friction in lower pair joints, a simple joint may be replaced by a kinematically equivalent compound joint. For instance, C. F. Chang, KUAS ME 14 國 立 高 雄 應 用 科 大 機 械 系 7
Antifriction Bearings C. F. Chang, KUAS ME 15 Some Higher Pair Joints C. F. Chang, KUAS ME 16 國 立 高 雄 應 用 科 大 機 械 系 8
1. Cylindrical Roller [ 圓 柱 形 滾 子, 滾 動 對 ] Name: 1. Cylindrical roller 2. Rolling Pair Connectivity (DOF) : 1 C. F. Chang, KUAS ME 17 Name: Cam Pair Connectivity (DOF) : 2 2. Cam Pair [ 凸 輪 對 ] C. F. Chang, KUAS ME 18 國 立 高 雄 應 用 科 大 機 械 系 9
Name: Rolling Ball Connectivity (DOF) : 3 3. Rolling Ball C. F. Chang, KUAS ME 19 Name: Ball in Cylinder Connectivity (DOF) : 3+1=4 4. Ball in Cylinder C. F. Chang, KUAS ME 20 國 立 高 雄 應 用 科 大 機 械 系 10
5. Spatial Point Contact Name: Spatial point contact Connectivity (DOF) : 3+2=5 C. F. Chang, KUAS ME 21 Replacement of a Higher Pair Joint with Lower Pair Joints In order to reduce the contact stress in higher pair joints, a joint may be replaced by some kinematically equivalent lower pair joints. For instance, a pin-in-a-slot joint may become a combination of a revolute joint and a prismatic joint. + C. F. Chang, KUAS ME 22 國 立 高 雄 應 用 科 大 機 械 系 11
Some Examples of Compound Joints C. F. Chang, KUAS ME 23 Mechanism & Linkage (p.8) A linkage is a closed kinematic chain with one link selected as the frame. A frame or base member is a link that is fixed. The term mechanism is somewhat interchangeable with linkage. In normal usage, mechanism is somewhat more generic term encompassing systems with higher pairs, or combinations of lower and higher pair joints, whereas the term linkage tends to be restricted to systems that have only lower pair joints. C. F. Chang, KUAS ME 24 國 立 高 雄 應 用 科 大 機 械 系 12
Planar Linkages A planar linkage is one in which the velocities of all points in all members are directed parallel to a plane, called the plane of motion. 機 構 上 各 點 之 速 度 若 皆 與 運 動 平 面 平 行, 則 稱 其 為 平 面 機 構 C. F. Chang, KUAS ME 25 Representation of Links and frame Binary links( 二 接 頭 桿 ) those that have two joints mounted on them Ternary links ( 三 接 頭 桿 ) those that have three joints mounted on them Slider-crank linkage Quaternary links ( 四 接 頭 桿 ) those that have four joints mounted on them C. F. Chang, KUAS ME 26 國 立 高 雄 應 用 科 大 機 械 系 13
Symbolic Designation of Single-Loop Linkages RRRR Linkage (4R) RPRP Linkage (2R-2P) RRRP Linkage (3R-P) C. F. Chang, KUAS ME 27 Visualization of the Motion of Linkages Modelling with woods, paper cards, Modelling with computer graphics systems C. F. Chang, KUAS ME 28 國 立 高 雄 應 用 科 大 機 械 系 14
Constraint Analysis of Planar Linkages(pp. 11-18) 18) Mobility (Degrees of freedom of a linkage) The minimum number of coordinates needed to specify the positions of all members of the mechanism 確 定 機 構 各 桿 件 之 相 對 位 置 所 需 之 獨 立 參 數 的 數 目 If the mobility is zero or negative, the assemblage is a structure. re. If the mobility is zero, the structure is statically determinate ( 靜 定 結 構 ) If the mobility is negative, the structure is statically indeterminate ( 靜 不 定 結 構 ) The mobility of planar linkages: (constraint criterion equation) n: the number of links M 桿 件 之 自 由 度 j: the number of joints j f i : the connectivity of joint i (dof of joint i) the dof a link with planar motion = 3 3( n 1) (3 ) i 1 f i M 3( n j 1) 接 頭 所 造 成 之 拘 束 度 j f i i 1 C. F. Chang, KUAS ME 29 Degree of Freedom of a Body (Link) The dof of a body is the number of independent coordinates needed to specify its position A body moving freely in a plane has three degrees of freedom. 2 translation + 1 rotation A body moving freely in space has six degrees of freedom. 3 translation + 3 rotation (pitch-yaw-roll) C. F. Chang, KUAS ME 30 國 立 高 雄 應 用 科 大 機 械 系 15
Examples Mobility analysis of a planar four-bar linkage M 3( n j 1) j f i i 1 Mobility analysis of a planar four-bar linkage C. F. Chang, KUAS ME 31 Examples(cont.) pp. 14-15 15 M 3( n j 1) when more than two members come together at a single point location (multiple joint 複 接 頭 ) n=6, j=7, f i =7 M=3(6-7-1)+7=1 j f i i 1 n=11, j=14, f i =15 M=3(11-14-1)+15=3 C. F. Chang, KUAS ME 32 國 立 高 雄 應 用 科 大 機 械 系 16
Remark on those linkages with all joints having connectivity one Since all joints having connectivity one (f i =1), we have f i =j=number of joints Moreover, if the mobility of planar linkages is set to one, the constraint criterion equation leads to 1=3(n-j-1)+j 3n=2j+4 n must be a even number, say n=2, 4, 6, M 3( n j 1) j f i i 1 C. F. Chang, KUAS ME 33 Constraint Analysis of Spatial Linkages(pp. 18-22) The dof of a link with spatial motion = 6 M 桿 件 之 自 由 度 接 頭 所 造 成 之 拘 束 度 6( n 1) i 1 6( n j 1) j (6 f ) j f i i 1 Where M = Mobility of spatial linkages n: the number of links j: the number of joints i f i : the connectivity of joint i (dof of joint i) This equation is known as the Kutzbach criterion C. F. Chang, KUAS ME 34 國 立 高 雄 應 用 科 大 機 械 系 17
Example 1 M 6( n j 1) j f i i 1 n = 4 ( 桿 數 ) j = 4 ( 接 頭 數 ) f i = 3+3+1+2 = 9 ( 接 頭 之 總 自 由 度 ) M = 6(4-4-1)+9 = -6+9=3 C. F. Chang, KUAS ME 35 Example 2 n = 7 j = 6 Five revolute joints: 1, 2, 4, 5, 6 One prismatic joint: 3 1 link joint f i = 5 1+1 1 = 6 ( 接 頭 之 總 自 由 度 ) M = 6(7-6-1)+6 = 6 C. F. Chang, KUAS ME 36 國 立 高 雄 應 用 科 大 機 械 系 18
n = 4 j = 4 (RSSR) Two revolute joints (f i = 1) Two spherical joint (f i = 3) Example 3 f i = 2 1+2 3 = 8 ( 接 頭 之 總 自 由 度 ) M = 6(4-4-1)+8 = -6+8 = 2 The result seems to conflict with our practical experience since there is a unique value of for any given value of. i.e., the orientation of link 4 can be determined when the orientation of link 2 is specified. Examining the mechanism carefully will reveal that we need an extra parameter to identify the orientation of link 3. Because this parameter doesn't affect the input-output relationship of the linkage, so we call it an idle degree of freedom. C. F. Chang, KUAS ME 37 Idle Degrees of Freedom (Redundant( DOF 多 餘 自 由 度 ) An idle dof is one that does not affect the input-output relationship of the linkage. Procedures for Locating the Idle dof are as following: Identify the input link and output link. Check to determine if a single link or a combination of connected links can move without altering the relative position of the input and output links. If the answer is positive, there are some idle dof s. C. F. Chang, KUAS ME 38 國 立 高 雄 應 用 科 大 機 械 系 19
Idle Degrees of Freedom & Stewart Platform For a Stewart platform, we have n = 14 (2 6 limbs+1 base link+1 output link ) j = 18 Six prismatic joints (f i = 1) Twelve spherical joint (f i = 3) f i = 6 1+12 3 = 42 ( 接 頭 之 總 自 由 度 ) M = 6(14-18-1)+42 = -30+42 = 12 Indeed, this mechanism has six idle dof. This is because each limb is free to spin about the line joining the centers of its spherical joints. C. F. Chang, KUAS ME 39 Planar Mechanism with an Idle Degrees of Freedom For the planar mechanism as shown in the figure, we have M = 1 if the kinematic pair at C is a rolling pair (f i =1) M = 2 if the kinematic pair at C is a cam pair (f i =2) However, the extra degree of freedom does not affect the the inputoutput (link6 vs. link 2) relationship of the linkage. So, the extra dof is an idle dof. C. F. Chang, KUAS ME 40 國 立 高 雄 應 用 科 大 機 械 系 20
Paradoxical Mechanism ( 矛 盾 機 構 ) ref pp. 25-29 29 over-constrained linkage A spatial 4R linkage is, in general, immovable because M=-2. However, it may have mobility one if special geometry are met. There are two well-know paradoxical mechanisms: Spherical four-bar mechanism (The axes of revolute joints all pass through a single point) Bennett mechanism a sin = b sin C. F. Chang, KUAS ME 41 Kinematic Inversion Kinematic Inversion is the transformation of one mechanism to another by choosing a different member to be the frame For example, Toothbrush mechanism Walking mechanism Water pump C. F. Chang, KUAS ME 42 國 立 高 雄 應 用 科 大 機 械 系 21
An Practical Application Water Pump C. F. Chang, KUAS ME 43 Classification of 4-bar4 Mechanisms & Grashof s s rule (pp. 32-37) 37) s: link length of the shortest link l: link length of the longest link p, q: link lengths of the other two links Type condition Shortest link mechanism Side link Crank-rocker Grashof s+l<p+q Coupler Double-rocker Base, frame Double-crank Change- Point s+l=p+q Any link Change-point Non-Grashof s+l>p+q Any link Triple-rocker Paper csme2001 csmmt2001 C. F. Chang, KUAS ME 44 國 立 高 雄 應 用 科 大 機 械 系 22
Example AB=1.14 in, BC=2.26 in, AD=1.74 in AF=2.00 in, DE=2.68 in, c=1.09 in Determine the region for joint E that will allow full rotation of link 6, i.e., EF=? Sol: Link AB in loop ABC can make a full rotation (BC-AB>c) Link AF is not the shortest one (AF<DE) Four-bar FEDA must be a crank-rocker s=ef l=de s + l < p + q EF+DE<AF+AD EF+2.68<2.00+1.74 1.74 2.0 2.68 E EF<1.06 in ANS C. F. Chang, KUAS ME 45 Analysis of four-bar linkages-centrodes C. F. Chang, KUAS ME 46 國 立 高 雄 應 用 科 大 機 械 系 23
Limit positions ( of Driven Link ) C. F. Chang, KUAS ME 47 Analysis of four-bar linkages-limit Positions ref: csme2001.pdf C. F. Chang, KUAS ME 48 國 立 高 雄 應 用 科 大 機 械 系 24
Classification of Spherical 4-bar4 Mechanisms Ref: csmmt2001 C. F. Chang, KUAS ME 49 Interference ref: csmeconf1995,1996,csmmtconf2000 CSMMTconf2000.pdf C. F. Chang, KUAS ME 50 國 立 高 雄 應 用 科 大 機 械 系 25
Actuators C. F. Chang, KUAS ME 51 Stable & Unstable Operation load > driving torque angular velocity is decreased until state A is reached End of Chapter 1 C. F. Chang, KUAS ME 52 國 立 高 雄 應 用 科 大 機 械 系 26