2. Fundamentals

7/26/2019 2. Fundamentals

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THEORY OF MACHINEWith:

Sami Salama Hussen Hajjaj


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How are we going to design?

2Sami Salama Hussen Hajjaj - May 2014

In order to design Machine Components of a given shape

under a known loading, we need to follow this procedure:

1. Analyze and Simplify Applied loadingMechanics & TOM

2. Identify the MaterialssStress-Strain limits Materials & MOM

3. Check if workpiece can support applied load

4. If yes, Select part that fits this requirements

5. If no, redo (what?)

MD


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CHAPTER 2:

FUNDAMENTALS


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Contents

Degree of Freedom (DOF)

Links, Joints, and Kinematic Chains

Kinematic Diagrams Inversion

Our Three Best Friends

The Grashof Condition

Mobility (DOF of the whole system)

(Half Joints vs. Full Joints)

4Sami Salama Hussen Hajjaj - Oct 2013


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DEGREE OF FREEDOM


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Degree of Freedom


DOF is the number of INDEPENDENT free motionsthat can begenerated by a joint or a linkage

DOF is the number of INDEPENDENT free parametersneeded to

describe the motions generated by a joint or a linkage


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Degree of Freedom


DOF is a very important concept is Mech Engineering

It is used in TOM, Machine Design, Robotics, and others

Links, Joints, Linkages, Machines, and Robots are classified

by their DOFs

Generally speaking, the higher the DOF, the more flexible

the design gets, but also the more complex it gets.


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LINKS, JOINTS, AND

KINEMATIC CHANES


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Links


Link: a rigid body used to link to other components of the machine

Each link possesses at least 2 nodes (Points for attachment to other links)

Links are classified according to # of nodes


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Links


Links can be any shape, but their classification still holds. Links are

classified according to # of nodes, not their shapes.

What are these links ?


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Joints


A jointjoins two 2 or more links at their nodes which will a

motion between the links

Joints can be classified based on:

DOF (Degree Of Freedom)

Order of Joints

Joint


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DoF Classification

Name Diagram DoF

Pin 1

Slider 1

Link against

plane

2

Pin in slot 2

Rolling

cylinder

Pure roll: 1

Pure slide: 1

Roll & slide: 2


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Order Classification

Name Diagram DoF ?

Single Joint

(1storder joint)

Higher order Joint

(2ndorder joint)

Order of Joint = Number of links joint1

A 2ndOrder Joint = 2 single joints = 2 DOF


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Links & Joints

Kinematic Chain

An assembly of links and joints

MechanismA kinematic chain designed for a purpose, AND have at

least one link grounded (Fixed to the ground)

Sami Salama Hussen Hajjaj - Oct 2013 14


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A Machine


AMachineis a system of multiple Mechanismsinteracting together toachieve a common purpose.


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KINEMATIC DIAGRAMS

17


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Kinematic Diagrams


Actual Kinematic Diagram

S i S l H H jj j O 2013 18


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Kinematic Diagrams


Actual Kinematic Diagram

S i S l H H jj j O t 2013 19


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Kinematic Diagrams



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S i S l H H jj j O t 2013 21


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Kinematic Diagrams


Sami Salama Hussen Hajjaj Oct 2013 22


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Kinematic Diagrams




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Kinematic Diagrams



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INVERSION



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Inversion


Inversion is created by grounding a different link in the

kinemetic chain.

This is usually done to see the impact of the ground(from different places) on other links

Refer to section 2.12 for more


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The 4 BarCrank Rocker

This is the FourBar Crank Rocker Mechanism. In the mechanism,

the crank rotates a full rotatation, while the rocker rocks back and

forth. And the coupler couples them


y

x

Crank

Coupler Rocker

Ground



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y

x

Slider

The 4 BarCrank Rocker

Crank

Ground

This is the FourBar Crank slider Mechanism. In the mechanism, the

rotaionl motion of the crank (full rotatation) is converted into a linear

horizontal motion, and the coupler couples them



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y

x

The 4 BarInverted Crank Slider

This is the FourBar Inverted Crank slider. It is very similar to the

crank slider, only that the slider is NOT grounded, therefore it is

alowed to slide along the coupler and rotate with the rocker.


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The Grashof Condition is a test used to predict the

behavior of the fourbar linkage, it is based on length of

each link

Given:S: length of shortestlink

L: length of longestlink

P: length of intermediatelink

Q: length of another intermediatelink

Therefore ..



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Case I: if S + L < P + Q, the fourbar behaves like a

Crank-Rocker (one link fully rotates, one rocks)

jj j

Grashof



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Case II: if S + L > P + Q, then the fourbar behaves like a

Rocker-Rocker (Double Rocker), none of the links make a

full revolution.

jj j

Non-Grashof



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Case III: if S + L = P + Q, then the fourbar behaves like

a Crank-Crank (Double Crank), both make full

revolution.

jj j

Grashof



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Example:

Calculate the Grashof condition of the fourbar mechanism

defined below:

A. 2 4.5 7 9

B. 2 3.5 7 9

C. 2 4.0 6 8



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Example:

Calculate the Grashof condition of the fourbar mechanism

defined below:

A. 2 4.5 7 9 => S+L< P+Q => Grashof (Crank-Rocker).

B. 2 3.5 7 9 => S+L > P+Q => non-Grashof (Double Rocker). C. 2 4.0 6 8 => S+L = P+Q => Grashof (Double Crank).


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MOBILITY



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Mobility

Mobility is the DOF of the whole mechanism (as a unit).

DOF => Joint

Mobility => Whole Mechanism

There are two kinds of Mobility:

Planar Mobility (2D Mechanisms)

Spatial Mobility (3D Mechanisms)

Our primary focus this semester will be on Planar Mobility



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Mobility

The meaning if Mobility:

Mobility = 0 ==> Rigid object

Mobility = 1 ==> all links and joints move in unison

(as one unit)

Mobility > 1 => 2 or more groups of lins/joints are able tomove independently from other groups



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Determining Mobility

Every component directly affects the mobility of the mechanism



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Solution for this example:

Number of Joints: 11.5

(11 Full Joints, 1 half Joint)

Number of Links: 9

Therefore:

M = 3(9-1)2(11.5) = 2423 = 1

Which means this system moves in unison (together)

Higher Order J

Half J



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In a 2D world, every link is able to make three motions.

Therefore: Every link adds 3 mobilities to the system



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If we add a joint to the link, then two motions are restricted.

Therefore: every joint removes 2 mobilities from the system



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If we add a joint to the link, then two motions are restricted.

Therefore: every joint removes 2 mobilities from the system



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And If we add a ground to the link (ground the link), then all

motions are restricted.

Therefore: every ground removes 3 mobilities from the system


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NO! .. some joints have 2 DOFs, they restrict only 1 motion.

Therefore:

- Joints with 1 DOF restrict only 2 motions

- Joints with 2 DOF restrict only 1 motion



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As such, Joints with 2 DOF restrict the amount of motions

restricted by 1 DOF joints, therefore

- Joints of 1 DOF (between 2 Links) are called: Full Joints

- Joints of 2 DOF (between 2 Links) are called: Half Joints



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Therefore, there are two approaches to calculating

mobility in a system:

One approach is to count each half joint as a (0.5 Joint)and simply add it to the numeber of joints in the system,

and then use the same equation as before

M = 3(L1)2J



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The second approach is to use the Kutzbachs/ Modified

Grueblersequation:

M= 3(L1)2J1J2

Where,

J1= Number of 1 DoF joints (Full Joints)

J2

= Number of 2 DoF joints (Half Joints)



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L (Gr)

L

L

L

L

L

JJ

J

JJ

1/2

J

J



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Higher order Joint vs. Joint

2 DOF, 2 Joints

2 DOF, Joint

To remove the confusion check the number of links

2 DOF & 2 Links = Joint

2 DOF & 3 Links = 2 Joints

3 Links

2 Links



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After counting links and Joints, we found:

Links = 6 links (inlcuding ground)

Joints = 7 Full Joints, and 1 half Joint

Lets calculate M using the (First Approach)

M = 3 (L- 1)2 (J)= 3 (6- 1)2 (7.5) = 1515 = 0

This system has 0 Mobility (Rigid)



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After counting links and Joints, we found:

Links = 6 links (inlcuding ground)

Joints = 7 Full Joints, and 1 half Joint

Now Lets calculate M using the (Second Approach)

M = 3 (L- 1)2 (J1

)(J2

)= 3 (6- 1)2 (7)(1) = 15141 = 0

This system still has 0 Mobility (Rigid)



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Kinematic Diagrams



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Solution for this example:

Number of Joints: 11.5

(11 Full Joints, 1 half Joint)

Number of Links: 9

Therefore:

M = 3(9-1)2(11.5) = 2423 = 1

Which means this system moves in unison (together)

Higher Order J

Half J



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