Analytical and Applied Kinematics Vito Moreno moreno@engr.uconn.edu 860-614-2365 (cell) http://www.engr.uconn.edu/~moreno Office EB1, hours Thursdays 10:00 to 5:00 1
This course introduces a unified and analytical approach to two (2) and three (3) dimensional kinematics and planar and spatial geometry and constraint motion. Applications to: mechanisms, robotics, biomechanics Some topics covered: Coordinate transformation operators Displacement operators Motion invariants Velocity and acceleration operators Link and joint constraints Analytical methods of mechanism synthesis and analysis Geometric error modeling Computational methods in kinematics and geometry 2
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Reference texts: Kinematics and Mechanisms Design, Suh,C.H. and Radcliffe, C.W., John Wiley and Sons, 1978. Mechanism Design, Erdman, A.G., Sandor, G.N., Kota, S., Prentice Hall, 4 th ed. 2001. Mechanism and Dynamics of Machinery, Mabie, H.H., Reinholtz, C.F., John Wiley and Sons, 4 th ed. 1987. Theoretical Kinematics, Bottema, O., Roth, B., Dover Publications, 1979. Introduction to Theoretical Kinematics, McCarthy, J.M. The MIT Press, 1990. 4
Basic definitions: Kinematics is part of Solid Mechanics Statics study of forces and moments apart from motion Dynamics Kinetics study of the action of forces and moments on the motion of bodies Kinematics Study of the relative motion apart from forces 5
Mechanism combination of several rigid bodies which are connected is such a way that relative motion between them is allowed. Function of a mechanism to transmit or transform motion from one rigid body to another (source to output). Types of mechanisms- Gear systems Cam systems Planar and spatial linkages 6
Gears Cams Planar and spatial linkages 7
Machine a mechanism or combination of mechanisms for the purpose of transferring force or motion. Motion Plane (2D) motion translation, rotation Spatial (3D) Motion Helical pitch rotation and translation Spherical all points at a fixed distance from a given point Cylindrical free rotation and translation along an axis 8
The link a solid (rigid) body which is connected to n other links Linkage links connected by joints Erdman and Sandor, Table 1.1 Planar Link Types 9
Joints Kinematic Pair (joint) = connection between two links which allows certain relative motion Lower pair relative motion described by single (1) coordinate e.g. revolute, prismatic, rolling pairs Higher pair relative motion >1 degree of freedom roll/slip, spherical ball and socket Kinematic Chain a set of links connected by joints 10
Lower Lower Suh & Radcliffe Fig 1.1 Kinematic Pairs Pg 4 11
Erdman and Sandor, Table 1.2 Dimensional Joints 12
Erdman and Sandor, Table 1.2 Dimensional Joints 13
Degree of Freedom no of independent parameters (input coordinates) to completely define the position of a rigid body Y Y A 2D 3 dof, 3D 6 dof B A Unconstrained rigid link Three independent variables X A YA X A X 14
Before joining, multiple links will have 3n DOF Y C Y B C D Y A A X A X X C X A YA X C YC 15
Connections between links result in loss of DOF Y C B D A ground X X A YA X C YC Pin joints loose 2 DOF, have only 1 DOF called f 1 joint Degree of Constraint = number of freedoms a free body looses after it is connected to a fixed link DOC+DOF=3 16
Four Bar Linkage Notation Input link A A o 2 fixed 1 D Coupler link 3 B o Mobility analysis by Gruebler s equation fixed 1 B DOF n 4 f # joint s with 1 DO relative freedom 1 Follower link F 3( n 1) 2 f # members 1 4 4 3(4 1) 2(4) 1 Four Bar Linkage is a Single DOF system -1 input coordinate required to define position of all members n f F F 1 17
Sliding connection reduces DOF also DOF F 3( n 1) 2 f y, z 0 n # members Input link A o A fixed 1 Slider-Crank Linkage Notation Coupler link 2 3 Output link f 1 4 # joint s include 1 B o Y B fixed pin joint s and 1 with 1 DO relative Freedom X 4 4 1 sliding joint s in Gruebler' s equation n f 1 F F 3(4 1) 2(4) D 18 f 1
N=12 F1=15 (12+3) F=3(12-1)-2(15)=+3 At Q, 3 links,2 joints 19
More complicated linkage (roll/slide) Velocity equivalent linkage (Kinematic diagram) 8 DOF= 3(n-1)-2f 1-1f 2 n=7 f 1 =7 f 2 =1 F=3(7-1)-2(7)-1(1)=+3 DOF= 3(n-1)-2f 1-1f 2 n=10 f 1 =12 f 2 =0 F=3(10-1)-2(12)-0(1)=+3 20
Exceptions to Grueblers equation n=5 f 1 =6 DOF 3(5-1)-2(6)=0 But motion is allowed 3 rd link is redundant Mfg errors could cause binding E&S Fig 1.26 Overconstrained linkage n=3 f 1 =3 DOF 3(3-1)-2(3)=0 But motion is allowed Sum of radii = dist between pivot points E&S Fig 1.27 21
Exceptions to Grueblers equation E&S Fig 1.26 Overconstrained linkage Passive or redundant DOF Rotation of 4 does not affect arm 3 n=4 f 1 =3, f 2 =1roll/slide DOF 3(4-1)-2(3)-1(1)=+2 Welded roller to arm(3) n=3, f 1 =2, f 2 =1 DOF=3(3-1)-2(2)-1(1)=+1 Slipping prevented between roller and cam n=4, f 1 =4, f 2 =0 DOF=3(4-1)-2(4)=+1 E&S Fig 1.28 22
6 Bar Linkages Tracer Points (binary links) Ternary links Watt Linkage E&S Fig 1.13 a-d Stephenson Linkage 23
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Changing the fixed Link - Inversion Basic Slider -Crank Whitworth Quick Return Link input output input output Oscillating Cylinder Pump Pump Mechanism input input output output 25
Force and Transmission of Motion n-n transmission of force and motion 26
Force and Transmission of Motion Transmission angle Pr essure angle 90 For maximum mechanical advantage 90 0 practically, 30 27
Homework #1 Determine the DOF for the Mechanisms shown 2 1 3 28
Homework #1 4 5 29
Homework #1 6 7 8 9 30
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Homework #1 11 Honda 32