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Kinematics Fundamentals

Chapter 2


Definitions • Mechanisms – A device which transform motion to some desirable pattern and typically develop very low forces and transmits little power

• Machine – Typically contains mechanism which are design to provide significant forces and transmit significant power


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Kinematics Fundamentals • Degree of Freedom (DOF) – The system’s DOF equal to the number of independent parameters(measurement) which are needed to uniquely define its position in space at any time


Kinematics Fundamentals

• Types of Motion –Pure translation –Pure rotation –Complex motion


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – A link is an rigid body which possesses at least two nodes which are points for attachment to other links


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – A joints (kinematic pairs) is a connection between two or more links, which allows some motion, or potential motion, between the connected links – Classification • Type of contact between the elements, line, point, or surface • Number of DOF allowed at the joint


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Classification • Type of physical closure of the joint • Number of links joined

– Type of Contact • Lower pair (full joints) – Describe joints with surface contact


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Type of Contact • Lower pair


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Type of Contact • Higher pair – Describe joints with point or line contact


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Number of DOF allowed Joint • One DOF (full joint)


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Number of DOF allowed Joint • Two DOF (half joint/roll-slide)


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Type of Physical Closure • Form closed- closed by its geometry • Force closed- closed by an external force


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Number of links joined • Order of the joint: the number of links minus one


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Kinematic Chain • An assemblage of links and joints, interconnected in a way to provide a controlled output motion in response to a supplied input motion

– Mechanism • A kinematic chain in which at least one link has been “grounded,” or attached, to the frame of reference


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Machine • A combination of resistant bodies arranged to compel the mechanical forces of nature to do work accompanied by determinate motions


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Ground • Any link or links that are fixed with respect to the reference frame

– Crank • A link which makes a complete revolution and is pivoted to ground


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Rocker • A link which has oscillatory (back and forth) rotation and pivoted to ground


Kinematics Fundamentals • Links, Joints, and Kinematic Chains – Coupler • A link which has complex motion and is pivoted to ground


Kinematics Fundamentals • Determining DOF – DOF or Mobility • The number of inputs which need to be provided in order to create a predictable output • The number of independent coordinates required to define its position

– Open or Closed – Dyads


Kinematics Fundamentals • Determining DOF – DOF in Planar Mechanisms • Gruebler’s Equation

M  3L  2 J  3G where – – – –

M = degree of freedom or mobility L = number of links J = number of joints G = number of grounded links


Kinematics Fundamentals • Determining DOF – DOF in Planar Mechanisms • Gruebler’s Equation

M  3L  2 J  3G • If more than one link is grounded, the net effect will be to create one larger, higherorder ground link. G is always one, therefore

M  3L  1  2 J


Kinematics Fundamentals • Determining DOF – DOF in Planar Mechanisms • Kutzbach Equation – Include full and half joints

M  3L  1  2 J1  J 2

where – – – –

M = degree of freedom or mobility L = number of links J1 = number of 1 DOF (full) joints J2 = number of 2 DOF (half) joints


Kinematics Fundamentals


Kinematics Fundamentals


Kinematics Fundamentals • Mechanisms and Structures – The DOF of an assembly of links completely predicts its character • If the DOF is positive→ mechanism • If the DOF is zero→ structure • If the DOF is negative → preloaded structure


Kinematics Fundamentals • Number Synthesis – The determination of the number and order of links and joints necessary to produce motion of a particular DOF


Kinematics Fundamentals • Paradoxes


Kinematics Fundamentals • Isomers


Kinematics Fundamentals • Linkage Transformation – Revolute joints in any loop can be replaced by prismatic joints with no change in DOF of the mechanism, provided that at least two revolute joints remain in the loop


Kinematics Fundamentals • Linkage Transformation – Any full joint can be replaced by a half joint, but this will increase the DOF by one – Removal of a link will reduce the DOF by one – The combination of rules 2 and 3 above will keep the original DOF unchanged


Kinematics Fundamentals • Linkage Transformation


Kinematics Fundamentals • Linkage Transformation – Any ternary or higher–order link can be partially shrunk to a lower–order link by coalescing nodes. This will create a multiple but will not change the DOF at the mechanism


Kinematics Fundamentals • Linkage Transformation – Complete shrinkage of a higher-order link is equivalent to its removal. A multiple joint will be created, and the DOF will be reduced


Kinematics Fundamentals • Intermittent Motion – Is a sequence of motions and dwells • Dwell; is a period in which the output link remains stationary while the input link continues to move • Geneva Mechanism


Kinematics Fundamentals • Intermittent Motion • Ratchet and Pawl


Kinematics Fundamentals • Intermittent Motion • Linear Geneva Mechanism


Kinematics Fundamentals • Inversion – An inversion is created by grounding a different link in the kinematic chain


Kinematics Fundamentals


Kinematics Fundamentals • Grashof Condition – Is a simple relationship that predicts the rotation behavior or rotatability of a four linkage’s inversion based only on the link lengths

S  L  PQ

• • • •

S = length of shorter link L=length of longest link P=length of one remaining link Q=length of the other remaining link


Kinematics Fundamentals • Grashof Condition – If the inequality is true, at least one link will be capable of making a full revolution with respect to the ground plane(Class I) – If not true, then the linkage is non-Grashof and no link will be capable of a complete revolution relative to any other link (Class II)

S  L  PQ


Kinematics Fundamentals • Grashof Condition – For the class I case: S+L< P+Q • Ground either adjacent to the shortest link and you get a crank-rocker • Ground the shortest link and you will get a double-crank • Ground the link opposite the shortest and you will get a Grashof double-rocker


Kinematics Fundamentals • Grashof Condition – For the Class II case: S+L> P+Q • All inversion will be triple-rockers in which no link can fully rotate

– For Class III: S+L=P+Q • All inversion will be either double-cranks, or crank-rocker


Kinematics Fundamentals


Kinematics Fundamentals


Kinematics Fundamentals


Kinematics Fundamentals • Classification of the Four Linkage – C. Barker developed a classification scheme that allows prediction of the type of motion that can be expected from a fourbar linkage based on the values of its link lengths – Link ratio formation – Letter designation (C), (R) - GCRR


Kinematics Fundamentals â&#x20AC;˘ Linkages of More Than Four Bars â&#x20AC;&#x201C; Geared Fivebar Linkages


Kinematics Fundamentals â&#x20AC;˘ Linkages of More Than Four Bars â&#x20AC;&#x201C; Sixbar Linkages


Kinematics Fundamentals • Spring as Links • Compliant Mechanism • Micro Electro-Mechanical Systems (MEMS)


Kinematics Fundamentals â&#x20AC;&#x201C; Problems


Kinematics Fundamentals â&#x20AC;&#x201C; Problems


Kinematics Fundamentals


Kinematics Fundamentals