Theory of Machines & MechanismsMCT 2212
Introduction Rigid Body Mechanics
Statics
Body at rest
Body with constant velocity
Dynamics
Body with accelerated motion
under the action of forces and moments
Theory of Machines and Mechanisms
Mechanisms/Linkages
Parts/ Link
Joints
deals with the determination of the forces and motions of links in machines
Subsystems of machines to facilitate analysis
Introduction
4
Links: rigid member having nodes, i.e. attachment points– Binary link: 2 nodes– Ternary link: 3 nodes– Quaternary link: 4 nodes
Links & joints
Joint: connection between two links (at their nodes) which allows motion
Classified by type of contact, number of DOF, type of physical closure, or number of links joined.
kinematic pair : Joints are also known as kinematic pair
5
Joint Classification
Type of contact:line/point i.e. higher pair, area/surface i.e. lower pair
Number of DOF: full joint=1DOF, half joint=2DOF
Form closed (closed by geometry) orForce closed (needs an external force to
keep it closed)Joint order = number of links-1
Full Joint: permits one relative motion between adjacent links. All of these
kinematic pairs are referred to as one degree of freedom(DOF) pairs.
Turning pairs allow relative turning motion between two
links., e.g. bearings, pivots, or pin joints.
Rolling pairs allow relative rolling motion between two links,
e.g. pair of friction wheels For a rolling pair, it is assumed that
there is no slippage between the links.
Sliding pairs allow relative sliding motion between two links,
e.g Piston-Cylinder.
Half Joint: allows two relative motions simultaneously
between the adjacent links and referred to as two degree of
freedom pairs.
Sliding pairs
Turning pairs
Half Joint
Kinematic Pairs
Types of joints
Higher Pairs & Lower Pairs
Lower pairs: A kinematic
pair or joint with
surface/area contact.
Higher Pairs & Lower Pairs
Higher pairs: A kinematic pair or joint with point contact or line contact.
Mechanism
Mobility: The mobility of a mechanism is defined as the number of independent parameters required to specify the position of all links of the mechanism. It also specify the number of input/actuators needed to operate the mechanism.
kinematic chain : A kinematic chain is an assembly of links formed by placing kinematic pairs at each of the nodes without specifying the ground link.Kinematic chains may be either open type or close type. Mechanism: It is an assemblage of links and joints with at least one link grounded and interconnected in a way to provide controlled output motions in response to supplied input motions.
DOF of a mechanism: The number of independent ways by which a dynamical system can move without violating any constraint imposed on it. In other words, the minimum number of independent coordinates, which can specify the position of the system completely. It is the number of parameters that determine the state of a physical system.
Link classification:
Ground: fixed wrt. reference frame
Crank: pivoted to ground, makes complete revolutions
Rocker: pivoted to ground, has oscillatory motion
Coupler: link has complex motion, not attached to ground
Machine: mechanism designed to do work.
A simple machine may also be considered as a single mechanism.
Figure 1.3(b) shows a free body diagram of the system used to analyze the manual force required to generate sufficient gripping force.
Figure 1.3(a) The tongs can be considered either as a machine or as a mechanism.
Figure: Inline 4-Cylinder EngineFigure: IC Engine Demonstration
Machine & Mechanism
Figure: A Paper/ Card Punching MachineFigure: Quick return mechanism
Machine & Mechanism
Figure: Slider crank mechanism
Figure: Scotch Yoke mechanism
Mechanisms are widely used in applications where precise relative movement and transmission of force are required. Motions may be continuous or intermittent, linear and/or angular.
Mechanisms
Worm_gear Gear-gear Gear-rack
Examples of continuous motion output
Examples of intermittent motion output
Cam –follower
Sewing machine creating a lockstitch using an
oscillating a boat shuttle
Sewing machine creating a lockstitch using Allen B Wilson's rotating hookGeneva
Mechanism
Cam –follower
Ratchet Mechanism
Every mechanism has one stationary base link. All other links may move
relative to the fixed base link. From the same kinematic chain, an inversion os
a mechanism is obtained by making the originally fixed link into a moving
link and selecting an originally moving link to be the fixed link .
Mechanism Inversion
Figure 1.39 Slider crank mechanism and its three inversions
(a) slider crank mechanism (link 1 fixed),
(b) inversion #1 (link 2 fixed),
(c) inversion #2 (link 3 fixed),
(d) inversion #3 (link 4 fixed).
Planar motion is restricted to a plane. For a planar mechanism, the motions of all of its links must take place either in the same plane or in planes that are parallel to one another. The slider crank mechanism and four-bar mechanism are examples of planar mechanisms.
Planar Mechanism
Figure 1.5 Slider crank mechanismFigure 1.7 Slider crank mechanism with offset
Figure 1.8 Four-bar mechanism
The Gruebler’s equation for the mobility, m, of a planar mechanism is given as
n= number of links in the mechanismJ1 =umber of one degree of freedom pairsJ2=umber of two degree of freedom pairs
212)1(3 JJnm
If, m< 0 i.e. “–ve”, Preloaded Structure, may require force to
assemble / Indeterminate problem .
If, m= 0 , Structure.
If, m>0 i.e. “+ve”, Mechanism.
Mobility
18Figure 1.36 Examples of mobility.
If, m< 0 i.e. “–ve”, Preloaded Structure, may require force to
assemble.
If, m= 0 , Structure.
If, m>0 i.e. “+ve”, Mechanism.
Mobility
19
Figure 1.36 (Continued)
n=6, j1=7, m=1
n=11, j1=14, j2=1 m=1
n=4, j1=1, j2=1, j3=2 ,m=3
n=5; J1=6; J2=0; m=0
n=5; J1=6; J2=0; m=0 but, m=1.Full Joint, Pure rolling, no sliding
In case of pure rolling,n=3; J1=3; J2=0; m=0
In case of rolling & sliding,n=3; J1=2; J2=1; m=1
Half Joint, rolling & sliding
3
1 11
24
5
1 1 1
2 4
5
3
23
1 1
Mobility Paradoxes
The Gruebler criterion pays no attention to
link sizes or shapes, it can give misleading
results in the face of unique geometric
configurations.
Paradoxical Mechanism
a sin= b sin
A spatial 4R linkage is, in general, immovable because M=-2.
However, it may have mobility one if special geometry are met.
There are two well-know paradoxical mechanisms:– Spherical four-bar mechanism (The axes of revolute joints allpass through a single point)– Bennett mechanism
Idle Degrees of Freedom
An Idle degree of freedom is one that appears (and is) present but its value has
no effect on the input – output relationships of interest
To identify Idle degrees of freedom, first identify the input and output links
–Then we must determine if a single link or combinations of links can move
without affecting the input/output link positions
–Like a connecting link rotating (about its axis) in a steering mechanism
without changing the relationship between the steering wheel and the front
tires in a vehicle
04/28/2023 ME 3230 Page 24
Note: Pin-in-slot & Cam Contact are half joints
52132)112(3
2)1(3213
12
21
2
1
JJnmJJn
Here,
The Structure has five DOF with two Idle DOF’s.
They are the roller and the cam rocker .
mActual = MTheoretical - mIdle
=5-2 = 3
Idle Degrees of Freedom
Joints: RSSRn=4, j1=2, j2=0, j3=2 ,m=2But Actual m=1
The result seems to conflict with our practical
experience since there is a unique value of for
any given value of . i.e., the orientation of link 4
can be determined when the orientation of link 2 is
specified.
Examining the mechanism carefully will reveal that
we need an extra parameter to identify the
orientation of link 3. Because this parameter
doesn't affect the input-output relationship of the
linkage, so we call it an idle degree of freedom.
Idle Degrees of Freedom (Redundant DOF)
An idle dof is one that does not affect the input-output relationship of
the linkage.
Procedures for Locating the Idle dof are
as following:
–Identify the input link and output link.
–Check to determine if a single link or a
combination of connected links can move
without altering the relative position of the
input and output links. If the answer is
positive, there are some idle dof’s Joints: n=14, j1=6, j2=0, j3=12 ,m=12But Actual m=6 and 6 idle dof, each idle dof locates in between two spherical joint.
Stewart Platform
Figure 1.29 (a) a prosthetic hand Figure 1.29(b) Fingers wrap around an object as shown in
In a spatial mechanism, links move in three dimensions. For example, in a prosthetic hand, the thumb moves in a plane that is not parallel to the planes of motion of the other four fingers.
where the subscript refers to the number of freedoms of the joint.
54321 2345)1(6 JJJJJnM
Spatial mechanism
The Kutzbach mobility equation for spatial linkages:
Example of Spatial Linkage
4 Links; 2 spherical Joints, 1cylindrical joint and 1 revolute joint.DOF of a Spherical Joint is 3DOF of a Cylindrical Joint is 2DOF of a Revolute Joint is 1
3231415)14(6
2345)1(6 54321
JJJJJnM
Four-Bar Mechanism-Grashof's Criterion
Four-bar mechanisms may be studied by distinguishing the link lengths as
follows:
s: the length of the shortest link
l: the length of the longest link
p, q: the lengths of the other two links
To assemble the kinematic chain it is necessary that,
lqps
The type of a four-bar mechanism may be determined using Grashof"s Criterion,
(i) (ii) (iii)qpls
Then, only case (i) offers all three types of a four-bar mechanisms.
qpls qpls
Class_I Class_II Class_III
(i) If s is the input link, then
the mechanism is a crank
rocker.
(ii) If s is the base link, then
the mechanism is a drag link.
(iii) If otherwise, then the
mechanism is a rocker-rocker.
Rocker_Rocker Change Point
qpls qpls qpls
Four-Bar Mechanism-Grashof's Criterion
Figure 1.43 Types of four-bar mechanisms (a) crank rocker, (b) drag link, (c) rocker-rocker.
For S+L<P+Q
Crank-rocker if either link adjacent to shortest is groundedDouble crank if shortest link is groundedDouble rocker if link opposite to shortest is grounded
For S+L>P+QAll inversions will be double rockersNo link can fully rotate
Figure: Four Bar double rockersFor S+L=P+Q (Special case Grashof)
All inversions will be double cranks or crank rockersLinkage can form parallelogram or antiparallelogramOften used to keep coupler parallel (drafting machine)
32
Parallelogram form Anti parallelogram form Deltoid form
Figure 1.47 Four-bar mechanisms:crank rocker
Let the lengths of the three moving links are r2= 2.0 cm; r3=4.0 cm; r4=5.0 cm, adjusting the length of the base link we can get the following inversion of four bar mechanism.