Today's topic is range of movement and rotatability. In this topic, we shall discuss the range

of movement that a link can perform in a particular linkage which obviously depends on the relative

link lines. More specifically, the important point in this area is the complete rotatability

of a particular link. Why this is important? Because, as we know most of the machines are

driven by an electric motor and the shaft of an electric motor is capable of unidirectional

complete rotation. There must be one link which is connected to this motor shaft should

be able to rotate completely, that is the presence of all other links should not prevent

it from undergoing complete or full rotation. This aspect of range of movement and rotatability

is most comprehensively studied for four-link mechanism. In this rotatability condition,

most important notion is known as Grashof's criterion.

Let us start with a 4R - linkage. That means, there are 4 links which are connected by four

revolute joins which we normally called a four-bar linkage. There are four kinematic

dimensions in such a 4R -- linkage, which we call the link-length. That is the distance

between successive kinematic pairs. Let us say, lmin is the shortest link-length

and lmax is the longest link-length, where as l prime and l double prime are the remaining

two link-lengths. Grashof's criterion says lmin plus lmax is less than the l prime plus

l two prime. If the chain satisfies this Grashof's criterion then we call it a Grashof chain.

In a Grashof chain, the shortest link can always make complete rotation with respect

to all other links, whereas the other three links that is except the shortest link can

only oscillate with respect to one another and this amount of oscillation is always less

than 180 degree. If in a Grashof chain the shortest link decides how the kinematic behavior

is?

In a Grashof linkage, we can see two crank-rocker linkages will be obtained if the shortest

link which is the crank is adjacent to the frame. That means a shortest link which has

two adjacent links. If I make any of this adjacent links as the frame, then the shortest

link will be able to make complete rotation with respect all other links that means, even

with respect to the frame. Thus the shortest link becomes the crank. Because there are

the two adjacent links to the shortest link it is connected on two sides to two different

links, any one of these if I keep fixed or keep it as the frame then we get two crank-rockers,

because all other links can only oscillate with respect to the frame.

If I make a kinematic inversion from a Grashof chain, we can also get a double-rocker linkage.

If the shortest link is the coupler, that is the link which is opposite to the shortest

link that means, which is not directly connected to the shortest link if I hold that link fixed

or use that as the frame then the coupler the shortest link undergoes complete rotation

but the links which are connected to the frame they can only oscillate so we get a double-rocker

mechanism. If I make another kinematic inversion from

the same Grashof chain, we can get a double-crank. A double-crank linkage will be obtained if

the shortest link is the frame, because all the other three links can rotate completely

with respect to the shortest link because, it is a relative motion that matters, if the

shortest link can make complete rotation with respect to all other links then all other

links can make complete rotation with respect to the shortest link. If the shortest link

is held fixed of the frame, then all the other three links can make complete rotation. We

get a double crank even the coupler is also capable of making complete rotation.

Let us look at the model of this 4R-chain where the red link is the shortest link that

is lmin and this green link is the longest link that is lmax. If the adjacent link to

the shortest link this yellow one is held fixed and this is a Grashof chain that is

lmin plus lmax is less than l prime plus l double prime. In this Grashof chain, if I

hold one of the links which is adjacent to the shortest links that is this yellow one

fixed, then we gets a crank-rocker because the shortest link is able to make complete

rotation with respect to this link, whereas this green link only oscillates with respect

to this link. We see that the red link is rotating completely, whereas the green link

is only performing oscillatory motion and we get a crank-rocker linkage. Another inversion,

if I hold this link adjacent to the shortest link held fixed, then also we will get a crank-rocker

because this link will also make complete rotation in that case and this green link

will only oscillate with respect to this link. Let us now use the same chain but with a kinematic

inversion. The shortest link is the coupler and the link opposite to the shortest link

that is this green link is held fixed. If I move this link we will see that these two

links which are input and output may be connected to the frame can perform only oscillatory

motion, whereas this coupler is undergoing complete rotation, because the shortest link

in a Grashof chain can always make complete rotation with respect to all the other links.

Let us see we start from this horizontal position for the coupler that is link 3. As we see,

it has rotated 90 degree in the clock wise direction, it has rotated 180 degree, it has

rotated 270 degree and it has performed complete rotation. From the same Grashof chain, by

making a kinematic inversion earlier we had a crank-rocker linkage and now we had a double-rocker

linkage. The thing to note is the extreme positions of various links, this is I call

link number 4, this is link 2, this is link number 3 and this is link number 1, which

is the fixed link. Let us see why link 4 oscillates? As we see,

during this movement link 2 and 3 have become collinear and this point cannot go any further

as a result link 4 now has to return. It cannot go in the clock wise direction anymore it

has to move anti-clock wise direction, so the link oscillates. Again in this position,

link 2 and link 3 have become collinear and this is the extreme position of link 4 it

has to now rotate clock wise. The same thing is true for link number 2 when link 3 and

4 become collinear link 2 occupies one of its extreme positions. The other extreme position

is again when link 2, link 3 and link 4 have become collinear and link 2 occupies an extreme

position. We note also this important thing which will be used later on for displacement

analysis that the extreme position of a link is taken, when the two other links become

collinear like this or like this . Let us now look at another kinematic inversion

from the same Grashof chain. Here, the red link that is the shortest link is held fixed

that is acting as the frame. All these three links can rotate completely with respect to

this shortest link. Consequently we will get, what is known as a double-crank linkage. That

means, all the links connected to the frame including even the coupler are able to make

complete rotation. We see that if it is a Grashof's chain, then by kinematic inversions

it is possible to get all the three possible varieties of linkages namely crank-rocker,

double-rocker and double-crank. Two of the inversions resulted in crank-rocker when the

shortest link is the frame. One inversion when the shortest link is the coupler we get

a double-crank and when the shortest link is the fixed link, we get a double-crank.

We see that, if it is a Grashof chain then by kinematic inversion we can get all the

three possible varieties of linkages namely double-crank, crank-rocker and double-rocker.

We get two crank-rockers if the shortest link is adjacent to the frame. We get a double-rocker,

when the shortest link is the coupler and we get a double-crank when the shortest link

is the frame. We have seen that in a Grashof chain it is the position of the shortest link

which decides the characteristics of the linkage, whether is a crank-rocker or double-crank

or double-rocker.

Let me now discuss in a Non-Grashof chain. Non-Grashof chain is defined as lmin plus

lmax is greater than l prime plus l two prime, where lmin is the shortest link-length and

lmax is the longest link-length and the other two links-lengths are l pirme and l two prime.

In this Non-Grashof chain whatever may be the inversion all links can only oscillate

with respect to one another.

In a Non-Grashof linkage, all four inversions are double-rocker linkages, because all the

links can only oscillate with respect to one another. However the angle of oscillation

can be more than 180 degree, whereas in a Grashof chain angle of oscillation was always

less than 180 degree. The rockers can cross the line of frame if it is a Non-Grashof linkage,

whereas in a Grashof linkage one can show that the rocker can never cross the line of

frame. If it remains above the line of frame in one configuration then during the entire

movement in a Grashof's double-rocker or the crank-rocker, the rocker can never cross the

line of frame, it will be either above the line of frame or below the line of frame.

Whereas in a Grashof linkage which is always a double-rocker, the rockers can cross the

line of frame that means, it can come from above to below or below to above of the line

of frame. We explain another thing that in a Non-Grashof

linkage there exists only one mode of assembly. The linkage can be driven from one configuration

to its mirror image configuration, this needs a further explanation as we shall do right

now. We have just now talked off mode of assembly,

let me explain what do we mean by mode of assembly?

For given link-lengths I can have a 4-bar linkage, in this configuration this O2, A,

B and O4. With the same link-lengths I could have also assembled this link just in its

mirror image configuration, which let me call O2, A prime, B prime, O4. This link-length

is exactly same as this link-length, this link-length is same as this link-length and

this link-length is same as this link length. O2, A, B, O4 and O2, A prime, B prime, O4

are two modes of assembly depending on whether assembled above the line of frame or below

the line of frame. It is needless to say that, these two configurations are mirror image

of each other with the mirror placed along this line of frame that is O2, O4.

If this chain is a Grashof linkage, then we can never drive from one assembly to the other.

That means, if I assembled it in this configuration then by driving this linkage that is by moving

this link I can never occupy this mirror image configuration. Whereas, in a Non-Grashof linkage,

if I start from one configuration then as I drive the linkage there will be an instant

where this mirror image configuration will be taken up by the linkage. These points will

be further explained with the help of models.

Let us now look at the model of this Non-Grashof chain. Here, the red link is the longest link

that is lmax, this yellow link is the shortest link that is lmin and these two are l prime

and l double prime the remaining two link-lengths. It is easily seen in this model lmin plus

lmax is more than the sum of l prime and l double prime. It is a Non-Grashof chain and

independent of which link we hold fixed, we always get a double rocker-linkage. In this

particular situation this longest link the red link has been held fixed.

We see, all the links are only performing oscillatory motion they are unable to make

complete rotation, consequently we get a double-rocker linkage. The thing to note, that in this Non-Grashof

double-rocker linkage this rocking links is crossing the line of frame that is this line.

It is now above, now it has crossed it is below the line of frame. It is true for this

rocker it was above the line of frame, in this configuration it is below the line of

frame. Whereas, in a Grashof double-rockers, the rockers could never cross the line of

frame, we could never get from above to below. Let me demonstrate that the mirror image configurations

are taken by the same assembly. Suppose, this is one assembly mode and we can imagine what

will be the mirror assembly mode that will be like this. If I drive this mechanism, one

can see that it has occupied the mirror image configuration. Thus, in a Non-Grashof linkage

one can drive the linkage from one mode of assembly to the other, which was not possible

in a Grashof linkage. That I will demonstrate with a different model.

Let us consider another kinematic inversion of the same Non-Grashof chain. If we remember

in the previous model this longest link was the frame. Here, the longest link has been

connected to the frame. This is the frame the fixed link. Here again, even in this inversion

we will get a double-rocker linkage. As we see that, this is one extreme position of

this red link and these two links have become collinear and this is another extreme position of this

red link when these two links have become collinear. We have also seen that both the

rockers could cross the line of frame. And one can easily see that mirror image configurations

are taken by the same mechanism, same assembly can be driven to the mirror image configuration.

For this mirror image configuration if we imagine will be something like this. The mechanism

could be driven from one mode of assembly to the other, as I said earlier is not possible

in a Grashof linkage. Let us now consider another inversion of the

same Non-Grashof chain. Here, as we see the longest link that is this red link is the

coupler, opposite to this link that is the frame. Even in this inversion this is a double

-rocker. The rocking angle is very large. But still all these links are unable to make

complete rotation. The nature of oscillation of this rockers, that is here as we see this

is crossing the line of frame in this direction, whereas this rocker is crossing the line of

frame in this direction, it is not crossing in this direction.

We see both these rockers are crossing the line of frame in the outward direction not

in this inward direction; such a rocker is called both -outward. Similarly, other inversions

from same chain though have double-rockers but the rocking movements are different either

inward-outward or both inward. That depends on the position of this longest link. With

the longest link as the coupler we got both outward oscillations. Whereas longest link

connected to the frame we will get inward-outward and longest link as the frame will get both

inward. We will see all the three models from the

same Non-Grashof chain, link-lengths in all these models are equal, only thing that longest

link is position differently, here it is frame, here it is coupler, here it is connected to

the frame and all these inversions have produced as we have seen are double-rocker linkages.

When these two links become collinear, this gets its extreme position. Similarly, when

these two links become collinear, this gets its one extreme position, same is true here

these two links are collinear and this gets its extreme position.

So we have seen different inversion from the same Non-Grashof chain always in double-rocker

linkages. As a result, such a Non-Grashof linkage is not much useful in real life, because

if it has to be driven by a motor then there has to be a crank, whereas no crank exists

in a Non-Grashof links. Thus, only the Grashof linkage is useful in practice if it has to

be driven by a motor, and the shortest link must be connected to the motor shaft.

Let me now go back to the model of a Grashof linkage and to show that in a Grashof linkage

one mode of assembly cannot be driven to the other mode of assembly which is the mirror

image configuration with the mirror placed along the line of frame. For example, this

is one mode of assembly. If we dismantle all these revolute joins, I could have assembled

it in the mirror image configuration with these two links vertical but below this line

of frame. As we see if we drive this mechanism, this line has become mirror image of its previous

configuration, but this link has not because, this is a Grashof linkage and if it is assembled

it one more, it can never be driven to the other mode of assembly. A Grashof linkage

has two distinct modes of assembly, whereas a Non-Grashof linkage has a single mode of

assembly. One more important thing is to see that, in a Grashof linkage it is the position

of the shortest link that decides the movement characteristics depending on where the shortest

link is it may be a crank-rocker, it may be a double rocker, it may be a double crank.

Whereas, in a Non-Grashof linkage it is always double-rocker independent of any kinematic

inversion, however it is the position of the longest link that decides the rocking characteristics

whether it will be both inward or both outward or inward-outward.

Let me now summarize what we have seen so far.

1. In a Non-Grashof linkage we have seen all four inversions are double-rocker linkages.

2. The angle of oscillation in a Non-Grashof linkage can be more than 180 degree.

3. The rockers can cross the line of frame. 4. There exists only one mode of assembly

the linkage can be driven from one configuration to its mirror image configuration.

5. The position of the longest link with respect to the frame decided the type of rocking movement

that is whether inward-outward or outward-outward or both outward or both inward or inward and

outward. For a Grashof linkage let me go through the

similar points. What we have seen, for a Grashof linkage all the three varieties of linkages

can be obtained from the same chain by kinematic inversion, two inversions give crank-rocker

linkages which are most useful, one inversion gives a double-rocker linkage and last inversion

gives a double-crank linkage. The angle of oscillation of the rocking links can never

be more than 180 degree, it has to be less than 180 degree. The rockers of a Grashof

linkage can never cross the line of frame. There exists two distinct modes of assembly

that is the two mirror image configuration and the linkage can never be driven from one

configuration to its mirror image configuration. And lastly, it is the position of the shortest

link with respect to the frame that decides the type of movement, that is if the shortest

link is frame then it is double crank, if the shortest link is the coupler then it is

double-rocker, if the shortest link is connected to the frame then it is a crank-rocker with

shortest link as the crank. We have done with both Grashof and Non-Grashof chain. Let me

talk of with boundary between Grashof and Non-Grashof which is known as transition linkage.

In the transition linkage, the sum of the longest link and the shortest link is exactly

equal to the sum of the remaining two links, that is lmin plus lmax is equal to l prime

plus l double prime. In general, a transition linkage behaves just like a Grashof linkage,

that is if the shortest link is the frame then we get a double crank, if the shortest

link is the coupler then we get a double-rocker, if the shortest link is connected to the frame

then we get crank-rocker. However, in this transition linkage where lmin plus lmax is

exactly equal to l prime plus l double prime then it is obvious that there will be configurations

when all the links become collinear. This collinear configuration is called uncertainty

configuration. In these transition linkages, there are configurations where all links become

collinear which are called uncertainty configurations. From these configurations, the linkage can

move in a non-unique fashion as we shall demonstrate later with a model.

We have discussed in a general what happens in a transition linkage but now we have to

discuss special cases of transition chain.

As we know the condition lmin plus lmax equal to l prime plus l double prime is also satisfied

with two pairs of equal links. That means there are two pairs, one pair of lmin and

the other pair is lmax. In this special case there are two varieties,

Case (i when the links of equal length are not adjacent, that means links of equal length

are opposite to each other when we call it a parallelogram chain. In a parallelogram

chain all four inversions are double-crank. So, all four inversions of parallelogram chain

in double-cranks linkages of course with uncertainty configuration where the parallelogram linkage

can flip into anti-parallelogram configuration as we shall see just now.

Let us now look at the model of this transition, linkage that is the special situation of the

transition linkage. Here these two links are of same lengths which are opposite to each

other and this coupler link is same as the frame link. That is, we have a pair of lmin

and a pair of lmax. However, because these two equal lengths are not connected directly

they are the opposite sides it forms a parallelogram and we call it a parallelogram linkage. As

this parallelogram linkage moves, it is easy to see that there will be instance where all

the 4 revolute pairs have become collinear. As a result, the linkage is passing through

its uncertainty configuration and from here the non-unique movement is possible. If sufficient

care is taken, we can maintain the parallelogram configuration. From this uncertainty configuration,

it can also flip back to anti-parallelogram configuration and it is no longer a parallelogram.

The two opposite sides are equal but it is in the closed configuration this is called

anti-parallelogram. At this uncertainty configuration, the linkage

becomes uncertain whether to maintain the parallelogram or to flip back into anti-parallelogram

configuration. This uncertainty configuration is true for all types of transition linkages

whenever lmin plus lmax is l prime plus l double prime. Again this is another uncertainty

configuration, we can either maintain the parallelogram or it can flip back to anti-parallelogram

configuration. To overcome this uncertainty configuration in a parallelogram linkage,

we can use an extra coupler a redundant coupler which we have seen earlier and I will show

it to you again.

Let us again look at this parallelogram linkage where this length is equal to this length

and this coupler length is equal to the frame length. This is a parallelogram linkage. This

parallelogram linkage has a redundant or extra coupler which is of same length as this original

coupler. As a result when these four revolute pairs become collinear apparently this parallelogram

linkage is passing through uncertainty configuration. This extra coupler which is not passing through

uncertainty configuration ensures that, the parallelogram is always maintained it can

never feed back to anti-parallelogram configuration. The parallelogram linkage is very useful because,

it maintains unit angular velocity ratio, this crank and the follower are always parallel,

so it transmit unit angular velocity ratio from the input to the output link. But to

ensure that it remains a parallelogram and it does not flip back to anti-parallelogram

configuration at the uncertainty configuration we must have this extra or redundant coupler.

Let me now summarize what we have just seen for a transition linkage with opposite sides

of equal link lengths.

Here as we see, the four revolute pairs namely O2, A, B and O4 have all become collinear.

As a result, all the links become collinear and from this configuration onwards the linkage

moves in a non-unique fashion. If O2A is driven in this direction, O4B can move in this direction

or can flip back in the opposite direction. If it moves in the same direction then it

maintains the parallelogram, whereas if it moves in the opposite direction then it flips

into the anti-parallelogram configuration.

Here we show that, the parallelogram linkage with revolute pair at O2, A, B and O4. What

see that, because this is a parallelogram this angle theta2 is always same as theta4

and it maintains unique angular velocity ratio between the input and the output link. At

the uncertainty configuration A, B prime O2, O4 everything becomes collinear.

And it can flip back into this anti-parallelogram configuration. O2A is still moving in the

counter-clock wise direction from the uncertainty configuration, whereas O4B has flip back and

moving in the clockwise direction and theta2 and theta4 prime that is the anti parallelogram

configuration they are not equal. Always the parallelogram configuration, that is to avoid

this anti- parallelogram configuration after crossing the uncertainty position we need

to have the extra redundant coupler as we explained with the help of a model.

Let us now discuss the second case of this special situation of a transition linkage

when we have two pairs of equal link-lengths.

Unlike in a parallelogram situation, here the links of equal length are adjacent, not

opposite to each other and this configuration where the links of equal lengths are adjacent

are called deltoid or kite configuration. From this deltoid or kite configuration, there

are two different possibilities. We will get a crank-rocker if any of the lmax that is

any of the longer links is held fixed and the connected lmin will be the crank. Whereas,

we get a double-crank if any of the lmin that any of the shortest links is held fixed. Such

a linkage when we have double-crank is called Galloway linkage. I will explain both this

deltoid configuration with the help of a model.

Let us look at one kinematic inversion from this kite configuration. Here as we see these

two links are of equal link-lengths and these two links is another pair of equal link-lengths.

Unlike in a parallelogram configuration here the equal link-lengths are adjacent to each

other rather than opposite of each other. These two links of equal lengths are adjacent,

these two links another pair of equal lengths are adjacent. So this is the kite configuration.

We are considering a kinematic inversion where one of the lmax that is one of the longest

links is held fixed. As a result we will get crank-rocker with the shorter link which is

connected to this fixed link will be the crank and the longer link will be the rocker. As

we see, we start from here, the shorter link can rotate completely whereas the longer link

is only oscillating. Here of course because it is a transitional linkage, there will be

uncertainty configuration when all the link-lengths become collinear. Here we see, there is loss

of unique movement the linkage can move like this which is no motion transmission or if

care is taken it can be driven as linkage with positive motion transmission. Here we

get a crank-rocker kinematic inversion with the longer link of this kite configuration

held fixed. Next we will see the model from the same chain where one of the shorter links

will be held fixed.

Let us now look at another kinematic inversion from the same kite configuration. Here again

one pair of longer links, one pair of shorter links. But one of the shorter links is held

fixed, previously we have seen one of the longer links which was held fixed. In this

kinematic inversion we will get a double-crank that means both the yellow link, the shorter

link and this red link will be able to make complete rotation. As we saw, both the red

link and the yellow were able to perform complete rotation. So this is a double crank.

There exists, a very fundamental difference between these double-crank and the double-crank

that we got earlier from a Grashof linkage or a parallelogram linkage. There one rotation

of the crank was also causing one full rotation of the follower. But here, we must have noticed

that it is two revolutions of the shorter crank, as we see the shorter crank has already

made one complete revolution, the longer crank is yet to make its complete revolution. If

I rotate the shorter crank one more revolution then the longer crank is completing its full

rotation. Thus, two revolutions of the shorter crank are generating one full revolution of

the longer crank. Such a mechanism is called a Galloway mechanism.

In fact, we can see that for this configuration of the shorter crank with the same link-lengths,

I could have added another configuration of this linkage. I can draw a circle with this

point as centre and this as radius, this point as centre and this as radius. These two circles

can intersect either here or at another point. Because two circles normally intersect at

two points. After one full revolution of the shorter crank, this point is going to the

other points of intersection of these two circles with this point as center and this

length as radius, this point as center and this as radius. It is a quite different type

of double-crank than the normal double-crank that we have encountered so far and this has

a special name as I said earlier is called a Galloway linkage.

Another trivial situation of a transition linkage occurs when all the link-lengths are

equal. That is lmin plus lmax is l prime plus l double prime, because all the four link

lengths are equal. With such equal link-lengths we get what is known as a rhombus linkage.

In a rhombus linkage, whatever may be the kinematic inversion, just like a parallelogram

linkage we will get double crank type linkages, of course only when uncertainty configurations

are avoided. Here again all the link lengths will become collinear at various configurations

and as we will see the linkage will move in an uncertain manner at this uncertainty configurations.

Let us now look at the model of this rhombus linkage. Here all the link-lengths are equal

that is this length is equal to the coupler link, is equal to the follower link and also

the frame link. All these four link lengths are equal as a result we get a rhombus. From

this rhombus linkage, all four kinematic inversions will give double prime, just like a parallelogram.

In this rhombus linkage also, as we see there are uncertainty configurations where all the

four revolute pairs become collinear and at these uncertainty configurations the linkage

moves in a non-unique fashion. If we maintain the rhombus it moves with a positive transmission

from input to the output, whereas at this uncertainty configuration that linkage moves

in a different way there is no transmission from this link 2 to link 4.

Again here, we get uncertainty configuration and there is no transmission from input to

the output link. However one can maintain the rhombus and get positive transmission.

This behavior is a very similar to the parallelogram linkage. We have discussed all types of 4R-linkages.

Let us see how we can extend Grashof's criteria that are Grashof like criteria for 3R-1P linkage.

Let us recall that a car slider is nothing but a revolute pair. Look at this figure we

have a revolute pair at O2, a revolute pair at A and a revolute pair at B and a car slider

between this link 4 and the fixed link that is link 1. If the centre of this circle of

this car slider is at O4 then this linkage is nothing but a 4R-linkage. With a revolute

pair at O2, A, B and O4, we can see the kinematic dimensions, l2 is a link-length which is obvious,

l3 is a link-length which is obvious and the other two links-lengths are O4B which we call

l4 and O2O4 which we call l1.

If we come to this 3R-1P linkage we have a revolute pair at O2, we have a revolute pair

at A and a revolute pair at B. Whereas between link 4 and 1 we have a horizontal prismatic

pair. We can imagine, this 3R-1P linkage is equivalent to having a revolute pair O4 at

infinity in a direction perpendicular to the direction of sliding which is horizontal.

We can think of a 4R-linkage O2, A, B and O4 where O4 is at infinity.

Let us look at the kinematic dimensions, here we have l2 the link-length O2A and l3 that

is the link-length AB, whereas the offset which is this e, that is the perpendicular

distance of O2 from the direction of relative sliding passing through B which is this line,

this we called offset which is e. Considering O4 at vertical infinity, say in this direction

or in this direction, because all these vertical lines meet at infinity then this e the offset

is standing out to be O2O4 minus O4B. If we call O2O4 as the l1 and O4B as l4 then this

offset is nothing but l1 minus l4. I could have considered this O4 at infinity in the

upward direction that is O4 is at vertical infinity in the upward direction. Then we

see that O2O4 which is l1 and it is this O4B which is l4 I would have got e equal to l4

minus l1. For 3R-1P mechanism I see there are two link-lengths

namely l1 and l4 which are infinite. Difference of these two infinities either l1 minus l4

or l4 minus l1 is the other kinematic dimension which we call offset e. We can write e as

the modulus of l1 minus l4 depending on whether I am considering O4B in the vertically upward

direction or vertically downward direction which will decide whether l4 is more than

l1 or l1 is more than l4, the difference of these two is the offset e. Keeping this in

mind, we can decide the Grashof like criterion, we see as we said e equal to l4 minus l1.

We say the Grashof condition turns out to be the shorter link-length lmin plus the longest

link-length l4 is less than the other two link-lengths that is the l1 plus the other

link-length which I for the time being, write l prime. l4 minus l1 is e this equation I

can write lmin plus e less than l prime. l4 the infinite link-length, l1 is the another

infinite link length but I have assumed l4 to be more than l1 so l4 becomes lmax. lmin

plus lmax less than l1 plus l prime is what we called equivalent Grashof's criteria for

3R-1P linkage and that I can convert to lmin plus e less than l prime, Where e is the amount

of offset lmin is the shorter link-length and l prime is the other link-length.

If the Grashof's condition is satisfied then the shorter link that is lmin can make complete

rotation with respect to all other links and we can get a slider crank mechanism. Whereas,

if it is a Non-Grashof slider crank that is lmin plus e is greater than the other link-length

l prime, then no link can make complete rotation and we will unable to get slider crank mechanism,

we will get a slider rocker mechanism. To conclude in today's lecture, what we have

seen the rotatability of 4R-linkage is most comprehensibly summarized by Grashof's criterion.

When we apply it to a 4R-linkage we have seen that, lmin plus lmax less than l prime plus

l double prime satisfies the Grashof's criteria and from a Grashof's linkage by kinematic

inversion we can get all kinds of linkages. Then we have seen the motion characteristics

of Non-Grashof linkage when Grashof condition is violated. We have also seen the boundary

between the Grashof and Non-Grashof linkage, which we called transition linkages. Then

we have seen special cases of transition linkages where the chain consists of two pairs of equal

link lengths. At the end, we have also seen how we can modify the Grashof's criterion

for a 3P-1P linkage and we got that lmin plus e less than l prime this is the equivalent

Grashof's condition for a 3 R1 P chain and if this Grashof's condition is satisfied then

this shortest link can make complete revolution with respect to all other link and the shortest

link can act as the crank of a slider crank mechanism.

I leave the students with a little problem, can we extend this Grashof criterion for an

R-R-P-P type 4R-links. There is a link between these two revolute pairs and there are links

between this revolute and prismatic, prismatic and prismatic and prismatic and this revolute.

If we recall, we had Scotch Yoke mechanism of this type and then elliptic trammel was

a mechanism of this type and Oldham's coupling is a mechanism of this type. In such a linkage,

as we see because there are prismatic pairs the link between connecting these R and P

pair is of infinite length, because the equivalent revolute pair corresponding to this prismatic

pair is at infinity. Similarly, this link which has both prismatic pair at its end is

also of infinite length and this link connecting P and R pair is also infinite link-length.

There is only one kinematic dimension, between these two revolute pairs let me call that

is l2. There is one kinematic dimension that is one link-length connecting two revolute

pairs and all other link-lengths are of infinite length. Consequently, Grashof's condition

is always satisfied as a result the shortest link l2 will be able to make complete rotation.

As we have seen, in the Scotch Yoke mechanism, the crank was always able to rotate completely

or the other two links we find in the elliptic trammel and Oldham's coupling. The shortest

link was able to make complete rotation with respect to all other links.