Intro to Kinematics of machines, DOF, Inversions

1. KINEMATICS OF MACHINES
UNIT 1

2. BOOKS TO READ
2
3
4
1

3. KINEMATICS OF MACHINES
• KINEMATICS: deals with only geometric aspects of motion without
any consideration of forces.
• MACHINE: A device for transferring and transforming motion & force
(power) from the source to the load(output).(eg. Shaper, screw jack)
• MECHANISM: Function of a mechanism is to transmit and modify a
motion.(eg. Clock, type writer)
• NOTE: Force is not greater than necessary to overcome friction b/w
moving parts in mechanism.
• For the study of kinematics, a machine may be referred to as a
mechanism, which is a combination of interconnected rigid bodies
capable of relative motion.

4. • Constrained Motion is defined as a mechanical pair which is having
definite motion with respect to another element.
• TYPES OF CONSTRAINED MOTION:
Irrespective of the force applied. Motion is possible in more than one
Direction & depends upon the direction
of the force applied.
Motion in one direction is stopped
by using some external means.

5. RIGID & RESISTANT BODIES
• RIGID BODY:
 Under the action of forces, it does not suffer from any distortion.
• RESISTANT BODY:
Those bodies which are rigid for the purposes they have to serve & capable
of transmitting the required forces with negligible deformation.
• A link or an element need not be a rigid body, but it must be a resistant
body. (semi rigid – belt is rigid in tension)
• Thus a link should have the following two characteristics:
1. It should have relative motion.
2. It must be a resistant body.
At this stage we will assume, the kinematic bodies are rigid & massless for
purposes of initial kinematic synthesis & analysis.

6. • KINEMATIC LINK/ELEMENT:
Resistant body or an assembly of resistant bodies which
go to make parts of a machine.
• KINEMATIC PAIR:
A pair is a joint of two links that permits relative motion.
• TYPES: (1. Nature of relative motion)
1 DOF
1 DOF
1 DOF
3 DOF
2 DOF
1 DOF
They should form one unit with no relative motion of parts with respect
to each other

7. • 2. Nature of contact
(a) Lower pair: A pair of links having surface or area contact b/w the
members.(sliding pairs, turning pairs & screw pairs)
(b) Higher pair: pair having a point or line contact b/w the links.
Nut & Screw Shaft rotating in a bearing Universal joint All pairs of mechanism
Cam & follower Roller bearings Gears
Low friction
High friction

8. • 3. Nature of mechanical constraint:
(a) when the elements of a pair are held together mechanically, it is
known as a closed pair. The contact b/w the two can be broken only by
destruction or damage of at least one part.
• All the lower pairs and some of the higher
pairs are closed pairs.
(b) When two links of a pair are in contact either due to
force of gravity or some spring action, they constitute an
unclosed/open pair.

9. TYPES OF JOINTS
• Binary joint: Two links are joined at the same connection.
• Ternary joint: Three links are joined at a connection.
1 Ternary joint equivalent to 2 Binary joints.
• Quaternary joint: Four links are joined at a connection.
1 Quaternary joint equivalent to 3 Binary joints.
• In general, if n number of links are connected
at a joint, it is equivalent to (n-1) binary joints.

10. DEGREES OF FREEDOM
• An unconstrained rigid body moving in space can be described by 3
translational motions & 3 rotational motions.(independent)
• Thus a rigid body possesses 6 DOF.
• The connection of a link with another imposes certain constraints on
their relative motion.
• Degrees of freedom of a pair is defined as the number
of independent relative motions, both translational &
rotational a pair can have.
DOF = 6 – No. of restraints
• Most practical mechanisms have 1 DOF.

11. KINEMATIC CHAIN
• A kinematic chain may be defined as a combination of kinematic
pairs, joined in such a way that each link forms a part of two pairs and
the relative motion between the links or elements is completely or
successfully constrained.
• For example slider crank mechanism(engine). The total combination
of these links is a kinematic chain.

12. In case the motion of a link
results in indefinite motion of
other links, it is a non- kinematic
chain
Redundant chain
J = Binary joints
L = Binary links
LHS>RHS – locked/redundant chain
LHS=RHS – Constrained
LHS<RHS – Unconstrained
For lower pairs

13. DEGREES OF FREEDOM OR MOBILITY:
• the number of inputs which need to be provided in
order to create a predictable output;
also:
• the number of independent coordinates required to
define its position.
Degree of Freedom in Planar Mechanisms:
Typical joints

14. MOBILITY OF MECHANISMS:
If one of the links of a kinematic chain is fixed to the ground
Space mechanisms
Planar mechanisms
Kutzbach’s criterion
Gruebler’s Criterion
• Applicable to linkages with 1 DOF.
• So for 1 DOF P2 = 0
Most of the mechanisms are expected
to have 1 DOF.
As P1 & N are to be whole no., the relation can
be satisfied only if N is even.
DOF Positive – Mechanism
DOF Zero – Structure
DOF Negative – Super/preloaded structure
N = 6 4B+ 2T or 5B + 1Q

15. PARADOXES
Because of unique geometry
Redundant link: which do not introduce any extra constraint
Redundant DOF:
One or more links of a mechanism can be moved without
causing any motion to the rest of the links of a mechanism.
Effective DOF
The function of the mechanism is not affected even if any
one of the links 2, 4 or 5 are removed. Thus, the effective no. of
links in this case is 4 with 4 turning pairs and thus has 1 DOF.

16. LINKAGE TRANSFORMATION
(No change in DOF)
There are several transformation techniques or rules that we can apply
to planar kinematic chains.
1. Revolute joints in any loop can be replaced by prismatic joints provided that
at least two revolute joints remain in the loop.
2. Any full joint can be replaced by a half joint, but this will increase the DOF by one.
3. Removal of a link will reduce the DOF by one.
4. The combination of rules 2 and 3 above will keep the original DOF unchanged.
Spring connection

17. COMMONLY USED LINKS & JOINTS
With 1 DOF, the mechanical press mechanism is constrained. Moving only one
link, the handle, precisely positions all other links in the press, sliding the press
head onto the work piece.

18. Determine the DOF of the mechanisms:
PROBLEMS
Take scale and red link to be
a single link attached by screw
Link 3 is Redundant DOF

23. • F = N – (2L + 1)
• P1 = N + (L – 1)
L = No. of loops in a mechanism
 Valid for mechanisms with turning pairs.
LOOP METHOD:

25. INTERMITTENT MOTION
• Intermittent motion is a sequence of motions & dwells.
• A dwell is a period in which the output link remains stationary while
the input link continues to move.
Geneva Mechanism Cam and follower Manufacturing systems

26. Expansion of pairs (Limit & disguise of revolute pairs)
During analysis of a mechanism, its pairs
may expand and the appearance of the
mechanism may change beyond recognition,
though the character of motion of kinematic
chain remains unaltered.
Inversions of Mechanism:
 Inversions are the different mechanisms obtained by fixing different links in a kinematic chain.
 Thus as many inversions are possible as the no. of links.
 The motion of a link in a kinematic chain relative to some other links is the property of the chain and not of the
mechanism.
 NOTE: Relative motion of links is not changed in any manner through the process of inversion.
The arrangement looks radically different from the original diagram but kinematically it is equivalent to the
Same 4 bar mechanism.(No relative motion is altered)

27. THE GRASHOF CONDITION
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 Grashof condition is a very simple relationship which predicts the rotation behavior or
rotatability of a four bar linkage's inversions based only on the link lengths.
S = length of shortest link
L = length of longest link
P = length of remaining link
Q = length of other remaining link
the linkage is Grashof and at least one link will be capable
of making a full revolution with respect to the ground plane.
This is called a Class I kinematic chain
If the inequality is not true, then the linkage is non-Grashof
and no link will be capable of a complete revolution relative
to any other link. This is a Class II kinematic chain.
• Crank
• Coupler
• Rocker
Impossible
d > a + b + c

28. • Whether they are later assembled into a kinematic chain in S, L, P, Q, or S, P, L, Q or any other order, will not change
the Grashof condition.
• The motions possible from a fourbar linkage will depend on both the Grashof condition and the inversion chosen.
The inversions will be defined with respect to the shortest link. The motions are:
For the Class II case, S + L > P + Q:
All inversions will be triple-rockers in which no link can
fully rotate.
For the Class I case, S + L < P + Q:

29. Referred to as special-case Grashof and also as a Class III kinematic chain, all inversions will be either
double-cranks or crank-rockers.
For the Class III case, S + L = P +Q
Both i/p & o/p rotate with same
angular velocities.
Both i/p & o/p rotate with different
angular velocities.
Wind shield wipers
i/p 2 rotations & o/p 1 rotation
(Galloway mechanism)

30. PROBLEMS
1. Find all the inversions of the chain given: 2. Indicate the type of mechanism whether
crank-rocker or double-crank or double-rocker

33. MECHANICAL ADVANTAGE
In general, the MA of a mechanism is defined as the ratio of the
force or torque exerted by the driven link to the necessary force or
torque required at the driver.
Both these angles are continuously changing and so is the MA.
When = 0o or 180o , linkage is said to be in toggle(or limit) posture.
If friction and inertia forces are ignored
The MA becomes infinite, thus at such a posture, only a small input
Torque is necessary to produce a very large output torque load.
Vise Grip locking pliers

34. TRANSMISSION ANGLE (µ)
• As this angle becomes smaller, the MA decreases and even a small amount of
friction might cause the mechanism to lock or jam.
• Ideally, we would like all of the force F34 to go into producing output torque T4
on link 4. However, only the tangential component creates torque on link 4.
The radial component F34 provides only tension or compression in link 4. This
radial component only increases pivot friction and does not contribute to the
output torque.
Therefore, the optimum value for the transmission angle is 90°. When µ is less
than 45° the radial component will be larger than the tangential component.
• A common thumb rule: µ > 45o or 50o
• The angle between the output link and the coupler is known as transmission angle.

35.  A double rocker 4 bar linkage has a dead-center posture
when links 3 & 4 lie along a straight line and the linkage is
locked. (µ = 0o or 180o)
 The designer must either avoid such a posture or provide
an external force, such as a spring to unlock the linkage.
 Therefore, the transmission angle has become a commonly
accepted measure of quality of a design of the 4 bar linkage.
Double crank or
Crank rocker
Double rocker

36. • It is important to realize that a toggle condition is only undesirable if it is preventing your linkage from
getting from one desired position to the other. In other circumstances the toggle is very useful. It can
provide a self-locking feature when a linkage is moved slightly beyond the toggle position and against
a fixed stop.
An example of such a toggle linkage is shown in Figure 3-2. It happens to be a special-case Grashof
linkage in the deltoid configuration which provides a locking toggle position when open, and folds on
top of itself when closed, to save space.

37. PROBLEMS

40. FOUR BAR MECHANISM INVERSIONS
Watt mechanism
Pantograph
Beam engine
Coupled locomotive
wheels

41. SLIDER CRANK MECHANISM INVERSIONS
First Inversion(Reciprocating engine and compressor)
Note that every mechanism has a fixed link called the frame. When different links are chosen as the frame,
the relative motions b/w the various links are not altered, but their absolute motions(those measured w.r.t
the frame) may be changed significantly.
For reciprocating engine, 4(piston) is the driver and
if it is a compressor, 2(crank) is the driver.

42. Second Inversion(Rotary engine & Whitworth quick return mechanism)

43. Third Inversion(Oscillating cylinder engine & Crank and slotted
lever mechanism)

44. Fourth Inversion(Hand Pump)

46. DOUBLE SLIDER CRANK CHAIN
First Inversion(Elliptical trammel)
Proof:

47. Second Inversion(scotch yoke)
• Used as valve control actuators
in high pressure pipelines.

48. Third inversion(Oldham coupling)
• Used to connect two parallel shafts when the
distance b/w their axes is small.

49. QUICK RETURN MECHANISMS
Many machine design applications have a need for a difference in average velocity between their
"forward" and "return" strokes. Typically some external work is being done by the linkage on the forward
stroke, and the return stroke needs to be accomplished as rapidly as possible so that a maximum of time
will be available for the working stroke. Many arrangements of links will provide this feature.
CRANK – SLIDER QUICK RETURN:
• Depending on the relative lengths of the links this mechanism is known as a Whitworth or crank slotted
mechanism. If the ground link is the shortest, then it will behave as a double-crank linkage or Whitworth
mechanism. If the driving crank is the shortest link, then it will behave as a crank-rocker linkage or crank
slotted mechanism.
• They are often used in metal shaper machines to provide a slow cutting stroke (due to reasons like heat
dissipation, life of tool bits etc.) and a quick return stroke when the tool is doing no work.

50. Whitworth mechanism Crank & slotted lever mechanism
Link 3 – uniform speed
Link 1 – variable angular velocity

52. In a crank and slotted lever quick-return mechanism shown, the distance between the fixed centres is 300
mm and the length of the driving crank is 150 mm. Find the inclination of the slotted lever with the vertical in
the extreme position and the ratio of time of cutting stroke to return stroke.

53. In a Whitworth quick return motion mechanism, as shown in
Fig., the distance between the fixed centres is 80 mm and the
length of the driving crank is 100 mm. The length of the slotted
lever is 180 mm
and the length of the connecting rod is 150 mm. Calculate the
ratio of the time of cutting to return strokes.
The distance between two parallel shafts connected by Oldham’s coupling is 25 mm. The driving shaft
revolves at 240 rpm. Determine the maximum speed of sliding of the tongue of the intermediate piece along
its groove.

54. In a crank and slotted lever mechanism, the length of crank is 560 mm and the ratio of time of working
stroke to return stroke is 2.8. Determine (a) distance between the fixed centres, and (b) the length of
the slotted lever, if length of stroke is 250 mm.

55. In a Whitworth quick return motion mechanism, as shown in Fig., the distance between the fixed
centres is 60 mm and the length of the driving crank is 80 mm. The length of the slotted lever is 160
mm
and length of the connecting nod is 140 mm. Find the ratio of the time of cutting stroke to the time of
return stroke and also the effective stroke.