ch 2 fundamentals lecture 2

7/23/2019 Ch 2 Fundamentals Lecture 2

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Ch 2: Fundamentals of Kinematics

•

Topic outline.• 2.1 Degrees of Freedom (DOF) or Mobility.

• 2.2 Types of motion.

• 2.3 Mechanism

• Kinematic Chain and linkage.

•Links and joints and their classification types.

• 2.4 Drawing kinematic diagrams. – Read on own.

• 2.5 Mobility and Connectivity.

• 2.6 Mechanisms and Structures.

• 2.8 Paradoxes.

• 2.9 Isomers.

• 2.11 Intermittent motion.

• 2.12 Inversions.

• 2.13 The Grashoff’s condition – four Bar and Rotatebility

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Degrees of Freedom (DOF)

• The number of independent parameters (measurements) required to uniquelydefine a mechanical system’s position in space at any instant of time.

• In this course a mechanical system will consist of rigid bodies

( no deformation ).

• For an elastic body (deformable) there are infinite degrees of freedom and is called

a compliant system.• A rigid body in plane motion has three DOF consisting of a translation ( x, y ) and

rotation (θ).

• A rigid body in three dimensional space has six DOF. 3 translation ( x , y , z) andthree rotation (α, β, γ ).

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Types of motion

•

General 3D motion:• Rotation about any axis and also simultaneous translation.

• Planar (2D) motion:

• Rotation about one axis and also simultaneous translation.

•

Pure rotation:• A center of rotation (point with no translation).

• All other points describe arcs about that center.

• Pure translation:

• All points on the body describe a parallel paths (curvilinear or rectilinear).

• A reference line on the body changes linear position and not angularorientation.

• Complex motion:

• A simultaneous combination of rotation and translation.

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Links

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Joints

• Kinematic pairs: A connection between two or more links at

their nodes, which allow relative motion in some directions

while constraining motion in others.

• Classification of joints:

1. Type of contact between the elements (lower or higher pair)

2. Number of DOF allowed at the joint (connectivity).

3. Type of physical closure of the joint (force or form closed).4. Number of links joined (order of the joint).

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Lower pair joints• Describes joints with surface contact.

• Six different types of lower pairs.

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Higher pair joints

•

Contact occurs only at isolated points or along line segments.• An infinite number of possible higher pair geometries.

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Additional joint type classifications

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Force closed

Form closed

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Kinematic chain, mechanism, and machine

• Kinematic chain: Assemblage of links and joints interconnected in a way

to provide a controlled output motion in response to a supplied input

motion.

•

Mechanism (or linkage): A closed kinematic chain in which at least onelink has been “grounded” or attached to a frame of reference. From

Chapter 1, a mechanism was defined as a system of elements arranged to

transmit motion in a predetermined fashion.

•Machine: – A collection of mechanisms arranged to transmit forces and do work.

– A combination of resistant bodies to compel the mechanical forces of nature

to do work accompanied by determinate motions.

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Some definitions• DOF of a single rigid body: Number of independent parameters required

to uniquely define the position of the body with respect to a given

reference frame.

• DOF of a system of rigid bodies: Number of independent parameters

required to uniquely define the position of all the members of the system.

• Connectivity: Number of DOF of a joint between two rigid bodies.

• Mobility: Number of DOF of a system of linkages.

• The mobility of a mechanism is the minimum number of coordinates

needed to specify the positions of all the members of the mechanism

relative to the frame.• The mobility of a mechanism is the number of inputs that need to be

provided in order to create a predictable output.

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Constraint Analysis

• Two independent members:

Each body has 3 DOF.

System DOF = 6.

• After forming a revolute joint:

DOF at the joint (connectivity) = 1.

System mobility reduced by 2.

Each 1-DOF joint reduces the mobility by 2.

•After forming a cam joint:

Connectivity = 2.

System mobility reduced by 1.

Each 2-DOF joint reduces the mobility by 1.

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Computation of Mobility

(Kutzbach’s modification of Gruebler’s criterion)

M = 3(L-1) – (2 J1 + J2)

• L = number of links.

• 1 = Ground frame link - loses all its mobility.

• J1 = Number of 1 DOF joints in the linkage formed.

• J2 = Number of 2 DOF joints in the linkage formed.

Simultaneously allows both translation and rotations

(half joint).

• Total reduction of mobility due to joints = 2J1 + J2

• System mobility = 3(L-1) – (2 J1 + J2)

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What if the mobility of a mechanism turns out to be

zero or negative

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(a)Mechanism. (b) A structure. (c) Preloaded structure.

statistically indeterminate

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1. Examples

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2. Examples

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3. Examples

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4. Examples

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1. Complex example

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2. Complex example

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3. Complex example

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1. Determine the mobility of the following

mechanism

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mechanism

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mechanism

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ParadoxesThe constraint criterion does not always predict the mobility of a linkage correctly

L = 5, J1=6, J2 = 0

M= 3(5-1)-2(6) = 0

Linkage has M = 1 only if all three

binary linkages are of equal lengths.

L = 3, J1=3, J2 = 0

M= 3(3-1)-2(3) = 0

Linkage has M = 1 because length of

Link 1 (distance between the 2

grounds) = lengths (radii) of links 2

and 3.

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Isomer

• Compounds that contain

the same number of

elements but differ in

arrangement and behave

quite differently.

• Linkages that contain the

same number of links but

differ in their structural

arrangement, and exhibit

different motion profiles.

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Isomer of mechanism with M =1

Number of valid

isomers

Links isomers

4 1

6 2

8 16

10 230

12 6856

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Fourbar nomenclature

Coupler link –

connected to both turning links.

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Turning link – Output

Connected to the base by

a revolute joint.

Turning

link

Ground link

A turning link can be either a crank or rocker:

• A crank is capable of a complete revolution about a fixed pivot on the base link.

• A rocker is only capable of oscillating between motion limits.

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InversionAn inversion is created by selecting a different link as the frame in the kinematic chain.

• Inversion of a slider-crank mechanism.

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Rotatability of fourbar mechanismThe motion possible from a fourbar linkage will depend on both the Grashof

condition and the inversion selected.

S = length of shortest link.L = length of longest link.

P, Q = lengths of the other two links.

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Grashof condition: S + L ≤ P + Q

Predicts the rotational behavior of a fourbar linkage

based solely on the link lengths.

S + L < P + Q Class I (Grashof)

Two joints on either side of

the shortest link perform

complete rotations and the

other two joints oscillatebetween specified limits.

One of the links undergoes

a complete revolution.

S + L > P + Q Class II (non-Grashof)

No joint is capable of

performing a complete

revolution. All joints

oscillate between specifiedlimits.

None of the joints

undergoes a complete

revolution

S + L = P + Q Class III (Special case Grashof

Behaves like on of the

Grashof sub-types but the

linkage behavior is

unpredictable.

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Motions from a Class I (Grashof) linkage

with inversions

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Crank-rocker

Double-rocker – s can make a

complete revolution

Double-crank

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Motions from a Class II (non-Grashof) linkage

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Triple-rockers: no link is capable of a full rotation

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Motions from a Class III (Sp. Case Grashof) linkage

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1: Determine the mobility, the Grashof condition, and the Barker

classification

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2. Determine the mobility, the Grashof condition, and the Barker

classification

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P2-43. Determine the mobility, the Grashof

condition, and the Barker classification

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End of lecture

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ch 2 fundamentals lecture 2

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