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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
,4
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.