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

Kinematics Introduction 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

Kinematics Introduction 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, 4th ed. 2001. Mechanism and Dynamics of Machinery, Mabie, H.H., Reinholtz, C.F., John Wiley and Sons, 4th ed. 1987. Theoretical Kinematics, Bottema, O., Roth, B., Dover Publications, 1979. Introduction to Theoretical Kinematics, McCarthy, J.M. The MIT Press, 1990.

Kinematics Introduction Basic definitions: Kinematics is part of Solid Mechanics Statics – study of forces and moments apart from motion Kinetics – study of the action of forces and moments on the motion of bodies Dynamics Kinematics – Study of the relative motion apart from forces

Kinematics Introduction 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

Kinematics Introduction Gears Cams Planar and spatial linkages

Kinematics Introduction 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

Kinematics Introduction 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

Kinematics Introduction 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 – more than one degree of freedom roll/slip, spherical ball and socket Kinematic Chain – a set of links connected by joints

Kinematics Introduction Suh & Radcliffe Fig 1.1 Kinematic Pairs Pg 4

Kinematics Introduction Degree of Freedom –no of independent parameters (input coordinates) to completely define the position of a rigid body 2D – 3 dof, 3D – 6 dof Y Unconstrained rigid link Three independent variables X

Kinematics Introduction Before joining, multiple links will have 3n DOF Y X

Kinematics Introduction Connections between links result in loss of DOF Y ground X Pin joints loose 2 DOF, have only 1 DOF called f1 joint Degree of Constraint = number of freedoms a free body looses after it is connected to a fixed link DOC+DOF=3

Erdman and Sandor, Table 1.2 Dimensional Joints

Erdman and Sandor, Table 1.2 Dimensional Joints

Kinematics Introduction Four Bar Linkage Notation Mobility analysis by Gruebler’s equation Coupler link 3 4 2 Input link Follower link fixed 1 fixed 1 Four Bar Linkage is a Single DOF system -1 input coordinate required to define position of all members

Kinematics Introduction Sliding connection reduces DOF also Coupler link 2 3 Input link Y fixed Output link 1 4 X Slider-Crank Linkage Notation 1 fixed

Kinematics Introduction F1=15 (12+3) F=3(12-1)-2(15)=+3 At Q, 3 links,2 joints

Kinematics Introduction Velocity equivalent linkage (Kinematic diagram) More complicated linkage 8 (roll/slide) DOF= 3(n-1)-2f1-1f2 n=7 f1=7 f2=1 F=3(7-1)-2(7)-1(1)=+3 DOF= 3(n-1)-2f1-1f2 n=10 f1=12 f2=0 F=3(10-1)-2(12)-0(1)=+3

Kinematics Introduction Exceptions to Grueblers equation n=5 f1=6 DOF 3(5-1)-2(6)=0 But motion is allowed 3rd link is redundant Mfg errors could cause binding E&S Fig 1.26 Overconstrained linkage n=3 f1=3 DOF 3(3-1)-2(3)=0 But motion is allowed Sum of radii = dist between pivot points E&S Fig 1.27

Kinematics Introduction Exceptions to Grueblers equation Passive or redundant DOF Rotation of 4 does not affect arm 3 n=4 f1=3, f2=1roll/slide DOF 3(4-1)-2(3)-1(1)=+2 Welded roller to arm(3) n=3, f1=2, f2=1 DOF=3(3-1)-2(2)-1(1)=+1 Slipping prevented between roller and cam n=4, f1=4, f2=0 DOF=3(4-1)-2(4)=+1 E&S Fig 1.26 Overconstrained linkage E&S Fig 1.28

Kinematics Introduction 6 Bar Linkages Tracer Points (binary links) Ternary links Watt Linkage Stephenson Linkage E&S Fig 1.13 a-d

Kinematics Introduction

Kinematics Introduction Changing the fixed Link - Inversion Basic Slider -Crank Whitworth Quick Return Link input input output output input Oscillating Cylinder Pump Pump Mechanism output input output

Kinematics Introduction – Force and Transmission of Motion n-n transmission of force and motion

Kinematics Introduction – Force and Transmission of Motion For maximum mechanical advantage

Kinematics Introduction – Homework #1 Determine the DOF for the Mechanisms shown 2 1 3

Kinematics Introduction – Homework #1 4 5

Kinematics Introduction – Homework #1 6 7 8 9

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Kinematics Introduction – Homework #1 11 Honda