![5. Four Bar Mechanism
ME 301 Theory of Machines I
Transmission Angle:
Alt[1] defined the transmission angle as:
𝑡𝑎𝑛𝜇 =
𝐹3
𝑑
𝐹3
𝑝 or 𝑠𝑖𝑛𝜇 =
𝐹3
𝑑
𝐹3
[1] Alt, Hermann (1889 - 1954). Der Übertragungswinkel und seine Bedeutung für das Konstruieren periodischer
Getriebe (The transmission angle and its importance for designing periodic mechanisms). Werkstattstechnik 26 (1932)
61–64.
3
A
A0
B0
a2
a4
2
4
a3 B
12 14
13
T12
T14
B
2F
2F+M
2F+M
F3∠13+
F3∠13+
F3∠13+
F3∠13
F3∠13
F3∠13
F3
d
F3
p
](https://image.slidesharecdn.com/week13-250422055131-5a58a0fc/85/Four-bar-Mechanical-engineering-branch-16-320.jpg)
Four bar Mechanical engineering branch .
- 1. 5. Four Bar Mechanism ME 301 Theory of Machines I A co-planar four-link mechanism with four revolute joints (R-R-R-R) is known as a four bar mechanism. Like the slider-crank mechanism and its kinematic inversions (R-R- R-P) it is one of the basic building blocks of many machines. After covering kinematic and force analysis in general we will concentrate on four-bar mechanisms.
- 2. 5. Four Bar Mechanism ME 301 Theory of Machines I All uses of a four-bar can be categorized in one of these three groups: 1. Correlation of Crank Angles / Function Generation 𝜃14 = 𝑓 𝜃12 By Ackermann.svg: User:Bromsklossderivative work: Andy Dingley (talk) - Ackermann.svg, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=11038290
- 3. 5. Four Bar Mechanism ME 301 Theory of Machines I All uses of a four-bar can be categorized in one of these three groups: 1. Correlation of Crank Angles / Function Generation 𝜃14 = 𝑓 𝜃12 𝑦 = 𝑥2 https://www.youtube.com/watch?v=HAdjYblt3hM
- 4. 5. Four Bar Mechanism ME 301 Theory of Machines I All uses of a four-bar can be categorized in one of these three groups: 1. Correlation of Crank Angles / Function Generation 𝜃14 = 𝑓 𝜃12 https://www.youtube.com/watch?v=noL5D4jLW6A
- 5. 5. Four Bar Mechanism ME 301 Theory of Machines I All uses of a four-bar can be categorized in one of these three groups: 2. Coupler point curve https://www.youtube.com/watch?v=4U8aREBFIDE
- 6. 5. Four Bar Mechanism ME 301 Theory of Machines I All uses of a four-bar can be categorized in one of these three groups: 2. Coupler point curve https://www.youtube.com/watch?v=NZZtXUcRZVs
- 7. 5. Four Bar Mechanism ME 301 Theory of Machines I All uses of a four-bar can be categorized in one of these three groups: 2. Coupler point curve https://www.youtube.com/watch?v=ESpEFJZp-co
- 8. 5. Four Bar Mechanism ME 301 Theory of Machines I All uses of a four-bar can be categorized in one of these three groups: 3. Positions of Coupler Link / Rigid Body Guidance https://www.youtube.com/watch?v=TLROAYXxkvA
- 9. 5. Four Bar Mechanism ME 301 Theory of Machines I Grashof’s Rule: The motion characteristics of a four-bar mechanism will depend on the ratio of link lengths. The links that are connected to the fixed link may have one of these two types of motion: • The link may have a full rotation around the fixed axis. This is called a crank (crank is also used for the input link) • The link may oscillate between two limiting angles. This is called a rocker.
- 10. 5. Four Bar Mechanism ME 301 Theory of Machines I Grashof’s Rule: A four-bar mechanism may have one of the following three types of motion: • Both links connected to the fixed link can have full rotation. This is called double crank or drag link mechanism. • Both links connected to the fixed link can oscillate between two limiting positions. This is called double rocker mechanism. • One of the links connected to the fixed link oscillates between two limiting positions while other can make full rotation. This is called crank rocker mechanism.
- 11. 5. Four Bar Mechanism ME 301 Theory of Machines I Grashof’s Rule: ℓ: Length of the longest link s: Length of the shortest link p, q: lengths of the two intermediate links 1. If ℓ + 𝑠 < 𝑝 + 𝑞 a. Two different crank-rocker mechanisms are possible. In either case shortest link is the crank and the fixed link is either of the adjacent links. b. One drag link (double crank) is possible when the shortest link is fixed. c. One double rocker mechanism is possible when the link opposite to the shortest is fixed.
- 12. 5. Four Bar Mechanism ME 301 Theory of Machines I Grashof’s Rule: 2. If ℓ + 𝑠 > 𝑝 + 𝑞 Only four different double rocker mechanisms are possible. 3. If ℓ + 𝑠 = 𝑝 + 𝑞 Same as case 1 however these mechanisms suffer from a condition known as change point position. At this position all the link center lines are collinear and this is a kinematically singular (undetermined) position. The follower at this position may rotate in either direction. The sign () that determines the closure is subject to change at this position.
- 13. 5. Four Bar Mechanism ME 301 Theory of Machines I Dead-Center Positions of a Crank Rocker Mechanism: Grashof’s Rule ℓ + 𝑠 < 𝑝 + 𝑞, 𝑠 is the crank a. Two different crank-rocker mechanisms are possible. In either case shortest link is the crank and the fixed link is either of the adjacent links.
- 14. 2. Kinematic Analysis Velocity and Acceleration Analyses Four-Bar Mechanism 𝑎2𝑒𝑖𝜃12 + 𝑎3𝑒𝑖𝜃13 = 𝑎1 + 𝑎4𝑒𝑖𝜃14 Re: 𝑎2𝑐𝑜𝑠𝜃12 + 𝑎3𝑐𝑜𝑠𝜃13 = 𝑎1 + 𝑎4𝑐𝑜𝑠𝜃14 1 Im: 𝑎2𝑠𝑖𝑛𝜃12 + 𝑎3𝑠𝑖𝑛𝜃13 = 𝑎4𝑠𝑖𝑛𝜃14 2 ሶ 1 : − ሶ 𝜃12𝑎2𝑠𝑖𝑛𝜃12 − ሶ 𝜃13𝑎3𝑠𝑖𝑛𝜃13 = − ሶ 𝜃14𝑎4𝑠𝑖𝑛𝜃14 ሶ 2 : ሶ 𝜃12𝑎2𝑐𝑜𝑠𝜃12 + ሶ 𝜃13𝑎3𝑐𝑜𝑠𝜃13 = ሶ 𝜃14𝑎4𝑐𝑜𝑠𝜃14 Let 𝜃12, ሶ 𝜃12 and ሷ 𝜃12 be the input: −𝑎3𝑠𝑖𝑛𝜃13 𝑎4𝑠𝑖𝑛𝜃14 𝑎3𝑐𝑜𝑠𝜃13 −𝑎4𝑐𝑜𝑠𝜃14 ሶ 𝜃13 ሶ 𝜃14 = 𝑎2𝑠𝑖𝑛𝜃12 −𝑎2𝑐𝑜𝑠𝜃12 ሶ 𝜃12 ሶ 𝜃14 = 𝑎2𝑠𝑖𝑛 𝜃12 − 𝜃13 𝑎4𝑠𝑖𝑛 𝜃14 − 𝜃13 ሶ 𝜃12 = 𝑔24 ሶ 𝜃12 ሶ 𝜃14 = 0 when 𝑠𝑖𝑛 𝜃12 − 𝜃13 = 0 → ቊ 𝜃12 − 𝜃13 = 0, Extended dead center 𝜃12 − 𝜃13 = 𝜋, Folded dead center ME 301 Theory of Machines I 2 3 4 A B3 A0 B0 B4 a2 a3 a4 12 13 14 a1
- 15. 5. Four Bar Mechanism ME 301 Theory of Machines I Dead-Center Positions of a Crank Rocker Mechanism: 𝜃12 − 𝜃13 = 0, Extended dead center 𝜃12 − 𝜃13 = 𝜋, Folded dead center
- 16. 5. Four Bar Mechanism ME 301 Theory of Machines I Transmission Angle: Alt[1] defined the transmission angle as: 𝑡𝑎𝑛𝜇 = 𝐹3 𝑑 𝐹3 𝑝 or 𝑠𝑖𝑛𝜇 = 𝐹3 𝑑 𝐹3 [1] Alt, Hermann (1889 - 1954). Der Übertragungswinkel und seine Bedeutung für das Konstruieren periodischer Getriebe (The transmission angle and its importance for designing periodic mechanisms). Werkstattstechnik 26 (1932) 61–64. 3 A A0 B0 a2 a4 2 4 a3 B 12 14 13 T12 T14 B 2F 2F+M 2F+M F3∠13+ F3∠13+ F3∠13+ F3∠13 F3∠13 F3∠13 F3 d F3 p
- 17. 2. Kinematic Analysis 2. Stepwise Solution of Loop Closure Equations Law of cosines: 𝑠2 = 𝑎1 2 + 𝑎2 2 − 2𝑎1𝑎1𝑐𝑜𝑠𝜃12 𝑠2 = 𝑎3 2 + 𝑎4 2 − 2𝑎3𝑎4𝑐𝑜𝑠𝜇 𝑐𝑜𝑠𝜇 = 𝑎3 2 + 𝑎4 2 − 𝑎1 2 − 𝑎2 2 + 2𝑎1𝑎2𝑐𝑜𝑠𝜃12 2𝑎3𝑎4 The extremums of the transmission angle is 𝑑𝜇 𝑑𝜃12 = 𝑠𝑖𝑛𝜃12 = 0 → ቊ 𝜃12 = 0 𝜃12 = 𝜋 ME 301 Theory of Machines I 2 3 4 A B A0 B0 a3 12 a2 a4 a1
- 18. 5. Four Bar Mechanism ME 301 Theory of Machines I Transmission Angle:
- 19. 5. Four Bar Mechanism ME 301 Theory of Machines I Mechanical Advantage: Definition: The mechanical advantage of a mechanism is the instantaneous ratio of output torque (force) to input torque (force). For a four bar mechanism where input is link 2 and output is link 4 𝑀𝐴 = 𝑇14 𝑇12 3 A A0 B0 a2 a4 2 4 a3 B 12 14 13 T12 T14 B 2F 2F+M 2F+M F3∠13+ F3∠13+ F3∠13+ F3∠13 F3∠13 F3∠13 F3 d F3 p
- 20. 5. Four Bar Mechanism ME 301 Theory of Machines I Mechanical Advantage: 𝑀𝐴 = 𝑇14 𝑇12 Neglecting friction, kinetic and gravitational potential energy changes of the links ℙ 𝑖𝑛 = ℙ 𝑜𝑢𝑡 −𝑇12𝜔12 = 𝑇14𝜔14 𝑀𝐴 = 𝑇14 𝑇12 = − 𝜔12 𝜔14
- 21. 5. Four Bar Mechanism ME 301 Theory of Machines I Mechanical Advantage: 𝑀𝐴 = 𝑇14 𝑇12 = − 𝜔12 𝜔14 = − ሶ 𝜃12 ሶ 𝜃14 = 𝑎4𝑠𝑖𝑛 𝜃14 − 𝜃13 𝑎2𝑠𝑖𝑛 𝜃12 − 𝜃13 𝑠𝑖𝑛 𝜃12 − 𝜃13 = 0, 𝑀𝐴 → ∞ Dead centers! 𝑠𝑖𝑛 𝜃14 − 𝜃13 = 0, 𝑀𝐴 = 0, 𝜇 = 0
- 22. 5. Four Bar Mechanism ME 301 Theory of Machines I Body Guidance-Two Position Synthesis: Chasles Theorem: The motion of a rigid body from one position to another in plane motion occurs most simply by a rotation about the pole P12 which is located at the perpendicular bisectors of two pairs of homologous points. Please note that when the separation between the two positions of the plane diminishes the homologous points define the velocity and the pole P12 boils down to instant center of zero velocity (ICZV).
- 23. 5. Four Bar Mechanism ME 301 Theory of Machines I Body Guidance-Two Position Synthesis: Chasles Theorem: