Dr. Samir Mustapha

Lecture 1

Mechanics of machines: MECH332

Some Terminology

What is a link?

What is a frame?

What is a joint or kinematic pair?

What are lower and higher pairs?

What is a closed loop kinematic chain?

What is an open loop kinematic chain?

What is a linkage?

What is planar motion, planar linkages?

What is spatial motion, spatial linkages?

What is inversion? Will be covered in the next lecture

What is a cycle and period?

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Mechanism and Machines

Machines: is a device for transferring and transforming

motion and force (power) from source to the load. It is also

referred as the study of the geometry of motion, and how

things move relative to each other.

Mechanism: is the division of machine design which is

related/concerned with the kinematic design of linkages,

cams, gears and gear trains.

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Terminology Links

The representation of a mechanism using lines is referred

to at the skeleton representation of the mechanism.

Lines connecting different parts on the mechanism are referred

to as Links:

Links are assumed to be rigid.

Links are joint together to form a kinematic chain.

Links are responsible to transmit motion and forces. 4

Type of links

Binary Link

Ternary Link

Quarternary Link

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Terminology - Frame

The fixed link is called a frame or a base link.

When there is no link that is actually fixed, we may consider

as being fixed and determine the motion of the links relative

to it. E.g. an automobile engine, the engine case/block may

be considered the frame, even though the automobile may

be moving.

Cylinder: frame

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Terminology Joints or Kinematic Pair

The parts that are connecting the links are known as

Joints (which permits constrained relative motion).

For instance, the joint between the crank shaft and the

connecting rod is refereed to a revolute joint or pin joint.

Revolute pair has 1 degree of freedom (DoF), one

element is fixed and the other is allowed to rotate.

DoF will be discuss in more depth later during the

course

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Terminology Lower and Higher Pairs

Connection between rigid bodies are classified as:

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Lower pair : the element of a lower pair are in surface

contact with one another. Lower pair include revolute or pin

connections (e.g. a shaft in a bearing). Only one DoF is

allowed.

Higher pair: the element of a higher pair are in point or line

contact with one another (if we disregard deflection). A

higher pair include a pair of gears or a disk cam and a

follower. More than one DoF is allowed (rolling and sliding )

Terminology Examples of Lower and Higher

Pairs

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Lower pair

Higher pair

Terminology Closed Loop Kinematic Chain

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Chain: A kinematic chain is an assembly of links and pairs

(joints).

Each link in a closed loop kinematic chain is connected to

two or more other links.

Lift platform

Terminology Open Loop Kinematic Chain

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One or more of the links is connected to one other link.

Industrial robot

Terminology: Linkage

Some references define linkages as kinematic chains

joined by only lower pairs.

The term is commonly used to identify any assemblage of

rigid bodies connected by kinematic joints. Linkage

configuration may serve as a component of a mechanism,

a machine or an engine.

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Terminology Planar Motion and Planar

Linkages

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If all points in a linkage move in a parallel planes, the

system undergoes planar motion and the linkage may be

described as planar linkage.

BMW F33 4 Series: four moving pieces swing into action, folding

one on top of the other until nothing is left of your trunk space.

Terminology Spatial Motion and Spatial

Linkages

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In a spatial mechanism, different particles may move in

paths which dont all always remain in a plane. This is

referred to as spatial motion and the linkage may be

described as spatial or three dimensional linkage (3D).

Boeing 707-338 series of aircraft Stewart platform

Symbols used for Kinematic diagrams

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Symbols used for Kinematic diagrams

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Mechanism and Machines: Mechanism

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Mechanism: Example 1

(Nutcracker)

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Mobility or Numbers of Degrees of Freedom

Mobility is one of the most fundamental concepts to the

study of Kinematics. The mobility of a mechanism is the

number of degrees of freedom it possesses. This

translates into a number of independent input motions

leading to a single follower motion. Equivalently it is the

minimum number of independent parameters required to

specify the location of every link within a mechanism. Oleg- p. 6

A single link constrained to move with planar motion has 3 DoFs. The x and y translation and the rotation angle.

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Mobility or Numbers of Degrees of Freedom

Two unconnected planar links six DoFs.

When the links are pinned together by a

revolute joint, the two-link system will

posses only four degrees of freedom.

Connecting two planar links with a

revolute joint had the effect of removing

two DoFs from the system. In other words,

revolute joint permits a single DoF

(rotation) between the connected links.

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Mobility : Grbler/Kutzbach Criterion

With this logic, it is possible to develop a general equation

which can help to predict the mobility of a planar

mechanism.

For instance, a planar mechanism with n links, each link will

have 3 DoFs. Recognising that one link is connected to the

ground, this will remove 3 DoFs, this will leave the system

with:

DoFsn )....1(3

212)1(3 ffnM

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Mobility : Grbler Criterion

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Each one degree of freedom joint removes 2 DoFs from the system, and similarly each two degrees of freedom joint removes 1 DoF. Then, the total mobility (M) of a system is given by Grublers equation: M= mobility of number of DoFs n= the total number of links (including the ground) f1= the number of 1 DoF joints f2= the number of 2 DoF joints

212)1(3 ffnM

Mobility : Grbler Criterion

Only four types of joints are commonly found in planar mechanisms: 1- Revolute (1 DoF) 2- Prismatic (1 DoF) 3- Rolling contact (1 DoF) 4- Cam or gear joint (2 DoF) The following definitions apply to the actual mobility of a device: : the device is a mechanism with M DoF (they are free to move once an external driving force is applied)

: the device is a statically determinate structure (constrained with no movement allowed, this forms a structure)

: the device is a statically indeterminate structure (this is overconstrained )

1M

1M

0M

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Mobility : GrblerKutzbach Criterion

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Grublers Criterion: Violation

Grublers formula does not take into account the specific

geometry of the mechanism (often occurs when several

links are parallel).

Consider the following examples:

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0

)6(2)15(3

M

M

This device can actually move

and is a mechanism with 1 DoF.

or

By definition, a pin joint connects two links.

When three links must be joined by 2 joints:

One connects link 2 to 3

The other connects link 3 to 4

Therefore all 3 links are joined.

4

3 2

2 3

4

Special case - mobility

Joint connecting k links at one joint must be counted as (k-1) joints

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Mobility Example 2

Determine the mobility of the device below.

There are four links and four revolute.

n= 4; the total number of links (including the ground)

f1= 4; the number of 1 DoF joints

f2= 0; the number of 2 DoF joints

This is 1 DoF mechanism.

1

089

0)4(2)14(3

M

M

M

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Mobility Example 3

Determine the mobility of the device below.

There are four links connected by 5 joints.

n= 5;

f1= 4; the joint connecting three links at a point count

twice

f2= 0;

This is a statically indeterminate

structure.

1

0109

0)5(2)14(3

M

M

M

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Mobility Example 4

Determine the mobility of the device below.

Three links exist, two 1 DoF revolute joint and one with 2 DoF

higher pair joint. In the higher pair, the two contacting bodies

may translate along the tangent line or rotate about the contact

point.

n= 3;

f1= 2;

f2= 1;

This is 1 DoF mechanism.

1

146

1)2(2)13(3

M

M

M

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Mechanism: Example 1

(Nutcracker)

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Determine the mobility of the device below.

n= 6;

f1= 7;

f2= 0;

This is 1 DoF mechanism.

1

)7(2)16(3

M

M

More Example - Grbler Criterion

Mobility of various

configurations of connected

links:

(a)n = 3, f1 = 3, f2 = 0, m = 0;

(b)n = 4, f1 = 4, f2 = 0, m = 1;

(c)n = 5, f1 = 5, f2 = 0, m = 2.

Effect of additional links on

mobility:

(a) m = 1, (b) m = 0 and (c)

m = -1.

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a

b

c

Mechanism and Machines: Mechanism

"Linkage mechanism simulator free software

available at : http://www.edu-ctr.pref.kanagawa.jp/LinkWeb/index_e.htm

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Mechanism and Machines: Machine

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"ROBOTEE LINKAGE 2.0 free software

available at : http://www.robotee.com/index.php/robotic-linkage-

mechanism-designer-and-simulator-linkage-2-0-all-free-81003/