1 Introduction gear

7/28/2019 1 Introduction gear

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Kinematics and Dynamics of Machines

1. Introduction

Instructor: S. Farhadi

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1.1. The Subject o f Kin ematics and Dyn amics of

Machines

The subject is a continuation of statics and dynamics

The objective ofkinematicsis to develop various means of

transforming motion to achieve a specific kind needed in

applications.

The objective ofdynamicsis analysis of the behavior of a

given machine or mechanism when subjected to dynamic

forces.

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1.2. Kinematics and Dynamics as Part of th e

Design Proc ess

The role of kinematics is to ensure the functionality of the

mechanism.

The role of dynamics is to verify the acceptability of induced

forces in parts

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1.3. Is It a Machin e, a Mechanism , or a Struc ture?

The term machineis usually applied to a complete product. A

caris a machine, as is a tractor, a combine, an earthmoving

machine, etc.

Each machine may have some devices performing specific

functions, like a windshield wiper in a car, which are called

mechanisms.

A st ructuredoes not have moving parts; its function is purely

structural, i.e., to maintain its form and shape under givenexternal loads, like a bridge, a building, or an antenna mast.


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1.4. Examples of Mechanisms; Terminolo gy

Punch Mechanism Skeleton representation of the

punch mechanism6

Windshield wiper mechanism

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Crank: A links which is able to make a complete revolution

and may be driven by a motor

Rocker: A links which is not able to make a complete

revolution

Coupler: A link which connects driver and follower

Driver: the input link

Follower: the output link

Frame (base link): The fixed link

Skeleton: A representation of the mechanism (replacing themembers with some links which connect essential points)

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Kinematic chain: an interconnected system of links in which

not a single link is fixed. Such a chain becomes a mechanism

when one of the links in the chain is fixed.

Planar mechanism: a mechanism in which all points move in

parallel planes

Compound mechanism:

Slider-crank mechanism:

Four-bar linkage:

Binary links: links with two connections to other links

Ternary link: links with three connections to other links.


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1.5. Mob il i ty of Mechan ism s

The mobi l i tyof a mechanism is its num ber of degrees of freedom. This

translates into a number of independent input motions leading to a single

follower motion.

A single unconstrained link has three DOF in planar motion.

If the two links are welded together, they form a single link having three

DOF.

A revolute joint in place of welding allows a motion of one link relative to

another. Thus, the two links connected by a revolute joint have four DOF.

A revolute joint eliminates two DOF.

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If the constraining condition allows only one DOF between the

two links, the corresponding joint is called a lower-pair joint.

If the constraint allows two DOF between the two links, the

corresponding joint is called a high-pair joint.

A low-pair joint reduces the mobility of a mechanism by two

DOF.

A high-pair joint reduces the mobility of a mechanism by one

DOF.

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Examples

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Kutzbachs criterion of mobility

m = 3(n 1) 2j1 j2

m: DOF

n: the number of links,

j1 : the number of low-pair joints

j2 : the number of high-pair joints.

Note that 1 is subtracted from n in the above equation to take

into account that the mobility ofthe frame is zero.


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Example: Mobility of various configurations of connected links:

(a) n = 3,j1 = 3,j2 = 0, m = 0;

(b) n = 4,j1 = 4,j2 = 0, m = 1;

(c) n = 4,j1 = 4,j2 = 0, m = 1;

(d) n = 5, j1 = 5,j2 = 0, m = 2.14

Effect of additional links on mobility

(a) m = 1,

(b) m = 0,

(c) m = -1.

When a structure has negative mobility, it is called an over-constrainedstructure.

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Kutzbachs formula for mechanism mobility does not take into

account the specific geometry of the mechanism, only the

connectivity of links and the type of connections (constraints).

Kutzbachs criterion can be violated due to the non-

uniqueness of geometry for a given connectivity of links.

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In compound mechanisms, there are links with more than two

joints. Kutzbachs criterion is applicable to such mechanisms

provided that a proper account of links and joints is made.

An example of a compound mechanism with coaxial joints at B.


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1.6. Kinematic Inversion

The process of choosing different links in the chain as frames

is known ask inemat ic invers ion

. In this way, for an n-linkchain n different mechanisms can be obtained.

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Four inversions of the slider-crank chain: (a) an internal combustion engine, (b) rotary

engine used in early aircraft, quick-return mechanism, (c) steam engine, crank-shaper

mechanism, (d) farm hand pump.

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1.7. Grashofs Law for a Four-Bar Linkage

Consider a four-bar linkage as presented. In this figure, s

identifies the smallest link, lis the longest link, andp, q are

two other links. Grashofs law, states that if the sum of the

shortest and longest links is not greater than the sum of the

remaining two links, at least one of the links will be revolving.

Grashofs law (condition) is expressed in the form:

s+ l=< p+ q

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Inversions of the four-bar linkage: (a) and (b) crank-rocker

mechanisms, (c) double-crank mechanism, (d) double-rocker

mechanism.


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Rotational Speed Ratio

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44

22

2

4

2

FP

PO

PO

EP=

==

SP

HO

FP

PO

GO

SP

PO

EP

4

4

4

44

2

2

22

2 ,

QO

QO

GO

HO

2

4

2

4

4

2==

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Constant Rotational Speed Ratio

In order to have a constant rotational speed ratio, the

transmitting line should intersect the center-points line in a

fixed point

This condition is valid for a wheel-belt mechanism, but is not

valid for a four-bar mechanism

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Sliding Contact

MPLPVS 42 =

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Rolling Contact

In order to have a pure rolling contact, the tangential

components of velocity for the contacting points have to be

equal and unidirectional. That happens solely when the

contact point lies on the centers point line.

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1 Introduction gear