Mechanics of Machines
Dr. Mohammad Kilani
Class 2
Fundamental Concepts
TYPES OF MOTION
Three Dimensional Motion
 A rigid body free to
move within a
reference frame will, in
the general case, have
a simultaneous
combination of
rotation and
translation.
 In three-dimensional
space, there may be
rotation about any axis
and translation that
can be resolved into
components along
three axes.
Plane Motion
 In a plane, or twodimensional space, rigid
body motion becomes a
combination of
simultaneous rotation about
one axis (perpendicular to
the plane) and also
translation resolved into
components along two axes
in the plane.
 Planar motion of a body
occurs when all the particles
of a rigid body move along
paths which are equidistant
from a fixed plane
Translation
 All points on the
body describe
parallel (curvilinear
or rectilinear) paths.
 A reference line
drawn on a body in
translation changes
its linear position but
does not change its
angular orientation.
Rectilinear Translation
Curvilinear Translation
Fixed Axis Rotation
 The body rotates about one axis
that has no motion with respect
to the “stationary” frame of
reference. All other points on the
body describe arcs about that
axis. A reference line drawn on
the body through the axis
changes only its angular
orientation.
 When a rigid body rotates about
a fixed axis, all the particles of the
body, except those which lie on
the axis of rotation, move along
circular paths
General Plane Motion
 When a body is
subjected to
general plane
motion, it
undergoes a
combination of
translation and
rotation, The
translation occurs
within a reference
plane, and the
rotation occurs
about an axis
perpendicular to
the reference
plane.
DEGREES OF FREEDOM (DOF) OR MOBILITY
Definition of the DOF
 The number of degrees
of freedom (DOF) that a
system possesses is
equal to the number of
independent parameters
(measurements) that are
needed to uniquely
define its position in
space at any instant of
time.
 Note that DOF is defined
with respect to a
selected frame of
reference.
xA
θB
xB
yA
YB
DOF of a Rigid Body in a 2D Plane
 If we constrain the pencil to always
remain in the plane of the paper,
three parameters are required to
completely define its position on
the paper, two linear coordinates (x,
y) to define the position of any one
point on the pencil and one angular
coordinate (θ) to define the angle of
the pencil with respect to the axes.
 The minimum number of
measurements needed to define its
position is shown in the figure as x,
y, and θ. This system o has three
DOF.
DOF of a Rigid Body in a 2D Plane
 Note that the particular parameters
chosen to define the position of the
pencil are not unique. A number of
alternate set of three parameters
could be used.
 There is an infinity of sets of
parameters possible, but in this
case there must be three
parameters per set, such as two
lengths and an angle, to define the
system’s position because a rigid
body in plane motion always has
three DOF.
DOF of a Rigid Body in 3D Space
 If the pencil is allowed to move in a
three-dimensional space, six
parameters will be needed to define
its position. A possible set of
parameters that could be used is
three coordinates of a selected
point, (x, y, z), plus three angles (θ,
φ, ρ).
 Any rigid body in a threedimensional space has six degrees
of freedom. Note that a rigid body
is defined as a body that is
incapable of deformation. The
distance between any two points on
a rigid body does not change as the
body moves.
ρ
θ
ϕ
DOF of Mechanisms
DOF of Mechanisms
DOF of Mechanisms
LINKS AND JOINTS
Links
 A link is a rigid body that
possesses at least two nodes
for attachment to other links.
 Binary link - one with two
nodes.
 Ternary link - one with
three nodes.
 Quaternary link - one with
four nodes.
Joints
 A joint is a connection between two or
more links (at their nodes), which allows
some motion, or potential motion,
between the connected links. Joints (also
called kinematic pairs) can be classified in
the following ways:
1. By the type of contact between the
elements, line, point, or surface.
2. By the number of degrees of freedom
allowed at the joint.
3. By the type of physical closure of the
joint: either force or form closed.
4. By the number of links joined (order of
the joint).
Joint Classification by Type of Contact
 The links joint by a joint may have a surface
contact (as with a pin surrounded by a hole), a
line contact (as with two cams), or a point
contact (as with a ball on a flat surface).
 the term lower pair describes joints with
surface contact. and the term higher pair to
describe joints with point or line contact.
 The main practical advantage of lower pairs
over higher pairs is their better ability to trap
lubricant between their enveloping surfaces.
This is especially true for the rotating pin joint.
A pin joint therefore is preferred for low wear
and long life, even over its lower pair cousin,
the prismatic or slider joint.
The Six Lower Pair Joints
Revolute (R) joint
Prismatic (P) joint
Helical (H) joint
Cylindrical (C) joint
Spherical (S) joint
Flat (F) joint
Joint Classification by Type of Contact:
Surface Contact (Lower Pairs)
 The pin joint or revolute
(R) joint and the translating
slider or prismatic (P) joints
are the only lower pairs
usable in a planar
mechanism.
Revolute (R) joint
Helical (H) joint
Spherical (S) joint
Prismatic (P) joint
Cylindrical (C) joint
Flat (F) joint
Joint Classification by Type of Contact:
Surface Contact (Lower Pairs)
 The screw or helical (H) joint,
the cylindrical (C) joint, the
spherical (S) joint, and flat (F)
joint are also lower pair joints
used in spatial (3-D)
mechanisms.
Revolute (R) joint
Prismatic (P) joint
 These joint pairs may be
obtained from a combination of
the R and P pairs.
Helical (H) joint
Spherical (S) joint
Cylindrical (C) joint
Flat (F) joint
Joint Classification by Type of Contact:
Surface Contact (Lower Pairs)
 In an (H) joint, motion of either
the nut or the screw with
respect to the other results in
helical motion.
 If the helix angle is made zero,
the nut rotates without
advancing and it becomes the
revolute (R) joint. If the helix
angle is made 90 degrees, the
nut will translate along the axis
of the screw, and it becomes
the prismatic (P) joint.
Revolute (R) joint
Helical (H) joint
Spherical (S) joint
Prismatic (P) joint
Cylindrical (C) joint
Flat (F) joint
Joint Classification by Number of Allowed DOF
 A more useful means to
classify joints is by the
number of degrees of
freedom that they allow
between the joint links.
 This is equal to the
number of independent
parameters that need to
be specified to
completely describe the
location of the one of the
links if the other link is
held fixed
Revolute (R) joint
Helical (H) joint
Spherical (S) joint
Prismatic (P) joint
Cylindrical (C) joint
Flat (F) joint
Joint Classification by Number of Allowed DOF:
One DOF Joints
 The pin joint or revolute (R) and
the translating (prismatic) slider
joint (P) are 1 DOF joints
because they allow only one
degree of freedom between the
joint links. These are also
referred to as full joints (i.e., full
= 1 DOF) and are lower pairs.
 The (R) and (P) joints are both
contained within (and each is a
limiting case of) the helical (H)
joint. The helical joint is
achieved by a screw and nut
arrangement.
Revolute (R) joint
1 DOF
Prismatic (P) joint
1 DOF
Helical (H) joint
1 DOF
Joint Classification by Number of Allowed DOF:
Two DOF Joints
 2 DOF Joints allow two
simultaneous
independent, relative
motions, between the
joined links.
 These joint are
sometimes referred to
as a “half joint.”
Example of these joints
are the cylindric (C)
lower pair joint, and the
pin in slot and the cam
roll-slide higher pair
joints.
Cylindrical (C) joint
2 DOF
Roll – Slide Cam joint
2 DOF
Roll – Slide Pin in Slot joint
2 DOF
Joint Classification by Number of Allowed DOF:
Two DOF Joints
 Note that if you do not allow the two
links in a roll-slide joint to slide, perhaps
by providing a high friction coefficient
between them, you can “lock out” the
translating (Δx) freedom and make it
behave as a full joint.
 This is then called a pure rolling joint and
has rotational freedom (Δθ) only. A
common example of this type of joint is
the automobile tire rolling against the
road.
 In normal use there is pure rolling and
no sliding at this joint. Friction
determines the actual number of
freedoms at this kind of joint. It can be
pure roll, pure slide, or roll-slide.
Roll – Slide Cam joint
2 DOF
Joint Classification by Number of Allowed DOF:
Three DOF Joints
 The flat (F) and the spherical, or balland-socket joint are examples of a
three-freedom joints. These two pairs
are lower pairs because they have
surface contact
 The flat joint allows two translational
and one angular independent
motions.
 The spherical joint allows three
independent angular motions
between the two links joined. This
joystick or ball joint is typically used
in a three-dimensional mechanism,
one example being the ball joints in
an automotive suspension system.
Flat (F) joint
3 DOF
Spherical (S) joint
3 DOF
Joint Classification by the type of physical closure
 A form-closed joint is kept together or closed by
its geometry. A pin in a hole or a slider in a twosided slot are form closed. In contrast, a forceclosed joint, such as a pin in a half-bearing or a
slider on a surface, requires some external force
to keep it together or closed.
 This force could be supplied by gravity, a spring,
or any external means. There can be substantial
differences in the behavior of a mechanism due
to the choice of force or form closure.
 The choice should be carefully considered. In
linkages, form closure is usually preferred, and it
is easy to accomplish. But for cam-follower
systems, force closure is often preferred.
Joint Classification by the number of links joined
(order of the joint)
 The simplest joint combination is
when two links are joint. This
produces a joint order of one.
Joint order is defined as the
number of links joined minus one.
As additional links are placed on
the same joint, the joint order is
increased on a one-for-one basis.
 Joint order has significance in the
proper determination of overall
degree of freedom for the
assembly.
KINEMATIC CHAINS,
MECHANISMS AND MACHINES
Kinematic Chains, Mechanisms and Machines
 A kinematic chain is defined as an
assemblage of links and joints,
interconnected in a way to provide a
controlled output motion in response to
a supplied input motion.
 A mechanism is defined as a kinematic
chain in which at least one link has been
“grounded,” or attached, to the frame of
reference (which itself may be in
motion).
Closed kinematic chain
 A machine is a collection of mechanisms
arranged to transmit forces and do work.
Closed kinematic chain
Open and Closed Mechanisms and Kinematic Chains
 Kinematic chains or mechanisms may
be either open or closed. A closed
mechanism will have no open
attachment points or nodes and may
have one or more degrees of freedom.
Open kinematic chain
Closed kinematic chain
 An open mechanism of more than one
link will always have one or more
degree of freedom, and requires as
many actuators (motors) as it has DOF.
A common example of an open
mechanism is an industrial robot.
 An open kinematic chain of two binary
links and one joint is called a dyad.
Closed Mechanism
Open Mechanism
Dyads
DETERMINING DEGREE OF FREEDOM OR
MOBILITY OF MECHANISMS
Degrees of Freedom of Planar Mechanisms
A. Mobility of one planar link = 3
B. Mobility of L planar links = 3L
C. Mobility of (B) when joint by J1 one DOF joints = 3L – 2J1
D. Mobility of (C) when joint by J2 two DOF joints = 3L – 2J1 – J2
E. Mobility of (D) with one grounded link = 3(L – 1) – 2J1 – J2
M = 3(L −1) − 2J1 − J2
where:
M = degree of freedom or mobility
L = number of links
J1 = number of 1 DOF (full) joints
J2 = number of 2 DOF (half) joints
Kutzbach’s Mobility Criterion for Planar Mechanisms
Degrees of Freedom of Spatial Mechanisms
M = 6(L −1) − 5J1 − 4J2 − 3J3 − 2J4 − J5
where:
M = degree of freedom or mobility
L = number of links
J1 = number of 1 DOF joints
J2 = number of 2 DOF joints
J3 = number of 3 DOF joints
J4 = number of 4 DOF joints
J5 = number of 5 DOF joints
Kutzbach’s Mobility Criterion for Spatial Mechanisms
Mechanisms and Structures
 The degree of freedom of an
assembly of links completely
predicts its character. There are
only three possibilities.
a) M > 0 → mechanism, links
will have relative motion.
b) M = 0, → structure, no
relative between links is
possible.
c) M < 0, → preloaded
structure, no relative
between links is possible
and some stresses may be
present.
L=4
J1 = 4
J2 = 0
M = 3(4-1) – 2×4 – 0 = 1
L=3
J1 = 3
J2 = 0
M = 3(3-1) – 2×3 – 0 = 0
L=2
J1 = 2
J2 = 0
M = 3(2-1) – 2×2 – 0 = -1
Example 1
1 (Ground)
2
3
4
5
6
L=9
J1 = 11
J2 = 1
7
9
DOF = 3(L-1) – 2J1 – J2
= 3×8 – 2×11 – 1
= 24 – 22 – 1
=1
8
Example 2
L=8
J1 = 10
J2 = 0
DOF = 3(L-1) – 2J1 – J2
= 3 × 7 – 2 × 10 – 0
=1
Assignment
Chapter 2
Problems: 2-8, 2-15 and 2-21
Paradoxes
 Because the Kutzbach’s criterion
pays no attention to link sizes or
shapes, it can give misleading
results in the face of unique
geometric configurations.
 The arrangement shown is known
as an “E-quintet,” and it has (DOF =
0) according to Kutzbach’s
criterion.
 Under certain link length
conditions, the constant distance
constraint imposed by one of the
links becomes redundant, and the
E-quintet becomes capable of 1
DOF motion.
L=5
J1 = 6
J2 = 0
M = 3(5-1) – 2×6 – 0 = 0
Paradoxes
 If no slip occurs in the cam
mechanism shown, Kutzbach’s
equation predicts zero DOF.
 If the two cams take the shape
of cylindrical disks, this linkage
does move (actual DOF = 1),
because the center distance, or
length of link the ground link, is
exactly equal to the sum of the
radii of the two wheels at any
time during the motion.
 The ground link constant length
constraint becomes redundant
if the two cams take the shape
of cylindrical disks, and pinned
on their respective centers.
L=3
J1 = 3
J2 = 0
M = 3(3-1) – 2×3 – 0 = 0
INVERSION
Inversion
 A mechanism was defined as a
kinematic chain with one of its links
grounded. An inversion of a mechanism
is obtained by releasing the grounded
link and grounding a different link from
the original kinematic chain.
 The number of possible inversions of a
mechanism is equal to its number of
links, and all inversions have the same
mobility or DOF.
FOUR BAR LINKAGE INVERSIONS
The Four Bar Linkage
 The four bar linkage is one of the simplest
mechanism for single-degree-of-freedom
controlled motion. It appears in various
disguises such as the slider-crank and the
cam-follower.
 It is in fact the most commonly used device
in machinery. It is also extremely versatile in
terms of the types of motion that it can
generate.
 Simplicity is one mark of good design. The
fewest parts that can do the job will usually
give the least expensive and most reliable
solution. Thus the four bar linkage should be
among the first solutions to motion control
problems to be investigated
The Four Bar Mechanism
 A four bar
mechanism is
obtained by
grounding one of
the links in the
four bar linkage.
 Four different
mechanism
inversions may be
obtained from the
same four bar
linkage, all with
DOF = 1
L=4
J1 = 4
J2 = 0
M = 3(4-1) – 2×4 – 0 = 1
Inversions of the Four Bar Mechanism
Crank Rocker 1 (GCRR)
Double Crank (GCCC)
Drag Link
Crank Rocker 2 (GCRR)
Double Rocker (GRCR)
The Grashof Condition on 4 Bar Linkage’s Rotatability
 The Grashof condition is a simple
relationship that predicts the rotation
behavior or rotatability of a fourbar
linkage based only on the link lengths.
P
Q
Let :
L = length of longest link
S = length of shortest link
P = length of one remaining link
Q = length of other remaining link
 In order for the crank to be pass through
point A without locking, the sum of the
lengths of the crank link and the ground
link (L+S) must be shorter that the sum of
the lengths of the two other links (P + Q).
S
A
L
S+L≤P+Q
The Grashof Condition on 4 Bar Linkage’s Rotatability
Let :
L = length of shortest link
S = length of longest link
P = length of one remaining link
Q = length of other remaining link
P
Q
S
L
Then if :
S+L≤P+Q
the linkage is Grashof class I linkage and
at least one link will be capable of making
a full revolution with respect to the
ground plane.
S+L≤P+Q
The Grashof Condition on 4 Bar Linkage’s Rotatability
 Based on the relationship between (S + L ) and (P + Q), the following Grashof
classes exist:
 (S + L < P + Q): Grashof Class I linkage.
At least one of the links is capable of making full rotation relative to the
other links
 (S + L > P + Q): Grashof Class 2 linkage.
None of the links is capable of making full rotation relative to the other
links.
 (S + L = P + Q): Grashof Class 3 linkage.
At least one of the links is capable of making full rotation relative to the
other links. will have “chang points” twice per revolution of the input
crank when the links all become colinear. At these change points the
output behavior will become indeterminate.
Motions of Grashof Class I Four Bar Linkage Inversions
S+L<P+Q
 The motions obtained from the
four inversions of of a Grashof
Class I four bar linkage, are as
follows
 Ground either link adjacent to
the shortest and you get a
crank-rocker.
 Ground the shortest link and
you get a double-crank
 Ground the link opposite the
shortest and you will get a
Grashof double-rocker, in
which both links pivoted to
ground oscillate and only the
coupler makes a full
revolution.
Motions of Grashof Class II Four Bar Linkage Inversions
S+L>P+Q
 None of
the links
can fully
rotate
relative to
an adjacent
link.
Triple Rocker #1
(RRR1)
Triple Rocker #2
(RRR2)
 All
inversions
will be
triplerockers
Triple Rocker #3
(RRR3)
Triple Rocker #4
(RRR4)
Motions of Grashof Class III Four Bar Linkage Inversions
S+L=P+Q
 All inversions will be either
double-cranks or crankrockers but will have
“change points” twice per
revolution of the input crank
when the links all become
colinear.
 At these change points the
output behavior will
become indeterminate. At
these colinear positions, the
linkage behavior is
unpredictable as it may
assume either of two
configurations.
Slider-Crank Inversions
Slider Crank
Crank Fixed
(Quick Return)
Coupler Fixed
(Crank-Shaper)
Slider Fixed
(Well Pump)
FOUR BAR LINKAGE TRANSFORMATIONS
Transformations of Four Bar Linkage
 The basic four bar linkage is a
loop of four links joint by four
revolute joints. If we relax the
constraint that restricted us to
only revolute joints, we can
transform this basic linkages to
a wider variety of mechanisms
with greater usefulness.
 There are several
transformation rules that we
can apply to planar kinematic
chains as discussed next
Transformations of Four Bar Linkage
Rule 1: Revolute Joints -> Prismatic Joints
 Revolute joints can be replaced
by prismatic joints with no
change in DOF, provided that
at least two revolute joints
remain in the loop.
Transformations of Four Bar Linkage
Rule 2: Full to Half Joints with Link Removal
 Any full joint can be replaced
by a half joint, but this will
increase the DOF by one.
 Removal of a link will reduce
the DOF by one.
 The combination of the two
rules above will keep the
original DOF unchanged.
INTERMITTENT MOTION MECHANISMS
Cam Follower Intermittent Motion Mechanisms
 Intermittent motion is a
sequence of motions and
dwells. A dwell is a period
in which the output link
remains stationary while
the input link continues to
move.
 There are many
applications in machinery
that require intermittent
motion. The cam-follower
variation on the four bar
linkage is often used in
these situations.
Geneva Mechanisms
 This is also a transformed four bar
linkage in which the coupler has
been replaced by a half joint.
 The input crank (link 2) is typically
motor driven at a constant speed.
The Geneva wheel is fitted with at
least three equispaced, radial slots.
The crank has a pin that enters a
radial slot and causes the
 Geneva wheel to turn through a
portion of a revolution. When the
pin leaves that slot, the Geneva
wheel remains stationary until the
pin enters the next slot. The result is
intermittent rotation of the Geneva
wheel.
Ratchet and Pawl
 The arm pivots about the center
of the toothed ratchet wheel and
is moved back and forth to index
the wheel. The driving pawl
rotates the ratchet wheel (or
ratchet) in the counterclockwise
direction and does no work on
the return (clockwise) trip.
 The locking pawl prevents the
ratchet from reversing direction
while the driving pawl returns.
Both pawls are usually springloaded against the ratchet. This
mechanism is widely used in
devices such as “ratchet”
wrenches, winches, etc.
Linear Geneva Mechanism
 This mechanism is analogous to an open Scotch yoke device with
multiple yokes and has linear translational output.
 It can be used as an intermittent conveyor drive with the slots
arranged along the conveyor chain or belt. It also can be used with a
reversing motor to get linear, reversing oscillation of a single slotted
output slider.