Chapter 2 – Kinematics Fundamentals 3/1/14 12:16 PM

2.1 – Degrees of Freedom or Mobility Mobility (M) – classified according to the number of degrees of

freedomo Degrees of Freedom - is equal to the number of

independent parameters (measurements) that are needed to uniquely define its position in space at any instant of time.

o Minimum number of DOF is three (x,y,z)o Any rigid body in a three-dimensional world has six degrees of

freedom (x,y,z) (theta, phi, rho)o Rigid body / link – is something that is incapable of

deformation. We will assume that kinematic bodies are rigid and massless

2.2 – Types of motion complex motion – simultaneous combination of rotation and

translationo Pure rotation – body possesses one point that has no motion

with respect to the “stationary” frame of reference. A reference line drawn on the body through the center changes only its angular orientation.

o Pure translation – All points on the body describe parallel paths. A reference line drawn on the body changes its linear position but does not change its angular orientation.

o Complex motion – Any reference line drawn on the body will change both its linear position and angular orientation. Points on the body will travel nonparallel paths, and there will be, at every instant, a center of rotation, which will continuously change location.

2.3 – Links, Joints, and Kinematic Chains Link – rigid body that possesses at least two nodes that points for

attachment to other linkso Binary Links – one with two nodeso Ternary Links – one with three nodeso Quaternary Link – one with four nodes

Joint – connection between two or more links which allows some motion or potential motion between connected links. Can be classified by:

o The type of contact between the elements, line, point, or surface

o The number of degrees of freedom allowed at the jointo The type of physical closure of the joint: either force or form

closedo The number of links joined (order of the joint)

Lower pair – to describe joints with surface contact ( pin surrounded by a hole)

Higher pair – to describe joints with point or line contact Classifying Joints

o One-Freedom – a rotating pin joint (R) and a translating slider joint (P). Also referred to as full joints and are lower pairs.

o Two-Freedom – sometimes referred to as half joint, with its two freedoms placed in the denominator. Also called the roll-slide joint because it allows rolling and sliding

o Three-Freedom – allows three independent angular motions between two links joined. Examples are a spherical or ball and socket joint. Typically used in three-dimensional mechanism. See pg. 34 for charts

o Pure Rolling Joint – has rotational freedom onlyo Friction determines the actual number of freedoms at this

kind of joint Pure roll, pure slide or roll-slide

o Form-Closed – joint is kept together or closed by its geometry

o Force-Closed – requires some external force to keep it together or closed

o Joint Order – the number of links joined minus one 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

Machine – a combination of resistant bodies arranged to compel the mechanical forces of nature to do work accompanied by determinate motions.

Crank – a link that makes a complete revolution and is pivoted to ground

Rocker – a link that has oscillatory (back and forth) rotation and is pivoted to ground

Coupler – a link that has complex motion and is not pivoted to ground

Ground – any link or links that are fixed (nonmoving) with respect to the reference frame.

2.4 – Drawing Kinematic Diagrams Page 36-37 - Drawings on Figure 2-4

2.5 – Determining Degree of Freedom or Mobility Dyad – an open kinematic chain of two binary links and one joint DOF in Planar Mechanisms

o Gruebler condition – used to predict the DOF of any assembly

M = 3L – 2J -3G where M = degree of freedom or mobility; L = number

of links; J = number of joints; G = number of grounded links

o Kutzbach’s modification of Grueblers equation M = 3(L-1) – 2J_(1) – J_(2)

J_(1) is the number of 1 DOF (full) joints J_(2) is the number of 2 DOF (half) joints

2.6 – Mechanisms and Structures If the DOF is positive, it will be a mechanism, and the links will

have relative motion If the DOF is negative then it is a preloaded structure, which

means that no motion is possible and some stresses may also be present at the time of assembly.

Structure – something with zero DOF. Example is three links joined by three full joints

Preloaded Structure – has a DOF of minus one. Example shows two links by two full joints.

2.7 – Number Synthesis Number Synthesis – the determination of the number and order of

links and joints necessary to produce motion of a particular DOF. Link Order - refers to the number of nodes (binary, ternary,

quaternary, etc) L = 4/ fourbar linkage Tabular approach pg. 45; also Table 2-2

2.8 – Paradoxes E-quintet is an example (Figure 2-10)

2.9 – Isomers Isomer means having equal parts. Depending on the particular

connections of available links, the assembly will have different motion properties.

2.10 – Linkage Transformation Transformation techniques or rules

o 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

o Any full joint can be replaced by a half joint, but this will increase the DOF by one.

o Removal of a link will reduce the DOF by one.o The combination of 2 and 3 above will keep the original DOF

unchangedo Any ternary or higher-order link can be partially “shrunk” to a

lower-order link by coalescing nodes. This will create a multiple joint but will not change the DOF of the mechanism.

o Complete shrinkage of a higher-order link is equivalent to its removal. A multiple joint will be created, and the DOF will be reduced.

2.11 – Intermittent Motion Intermittent Motion – a sequence of motions and dwells

o Dwell – a period in which the output link remains stationary while the input link continues to move’

Geneva Mechanism – A transformed fourbar linkage in which the coupler has been replaced by a half joint.

o Geneva Wheel – fitted with at least three equispaced, radial slots. The crank has a pin that enters a radial slot and causes the Geneva Wheel to turn through a portion of a revolution. When the pin leaves that slot, the Geneva wheel remains stationary until the pin enters the next slot.

o Ratchet and Pawl – the arm pivots about the center of the toothed ratchet wheel and is moved back and forth to index the wheel. The driving pawl rotates the ratchet wheel in the counter-clockwise direction and does no work on the return trip. The locking pawl prevents the ratchet from reversing direction while the driving pawl returns.

o Linear Geneva Mechanism – This mechanism is used as an intermittent conveyor drive with the slots arranged along the conveyor chain or belt.

2.12 – Inversion Inversion – created by grounding a different link in the kinematic

chain. Distinct Inversion – inversions that have distinctly different

motions2.13 – The Grashof Condition

Fourbar Linkage – simplest possible pin-jointed mechanism for single degree of freedom controlled motion.

3/1/14 12:16 PM

3/1/14 12:16 PM