Dynamics of Machines

Prof.Amitabha Ghosh
Department of Mechanical Engineering
Indian Institute of Technology, Kanpur
Module No. # 01
Lecture No. # 01
Rigid Body Motion
From the brief introduction to this subject, Dynamics of Machines, we have seen that
machines are primarily interconnected rigid bodies. If you have to study the Dynamics of
Machine, the first thing we have to understand that how rigid bodies behave under load
or under forces. That is, studying the motion of rigid bodies forms the first taped words
of the subject, Dynamics of Machines. We start today studying the motion of the rigid
body.
(Refer Slide Time: 00:58)

Rigid body motion can be studied under two headings: the first; the simpler case, which
you call plane motion and the more complicated type of rigid body motion, you call
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space motion. When do you call the motion of a rigid body to be a plane motion? It is the
situation where all the particles of the rigid body move in different directions no doubt,
but at any instant of time, all the velocity or displacement vectors lie in one plane or in
parallel planes.
On the other hand, in case of a space motion of a rigid body the velocity vectors of
different points will not lie in different directions or lie in the same plane or parallel
planes. So, studying plane motion is much simpler because you can project the whole
motion parameter or all the motion parameters like velocities, displacements and
accelerations on a projecting plane or the plane of the paper.
Next, let us see again, what are the different possible types you can have in the plane
motion. Let us first study plane motion.
(Refer Slide Time: 02:50)

What are the typical examples, say a wheel rolling on a horizontal surface here you know
that the velocities of different points will be different, but they will be all lying in parallel
planes. Similarly, we can have another example. If you hinge an object and suspend it by
two equal links, then again we know that this object will undergo a motion which you
can call plane motion, but again there is a subtle difference between these.
I will take another simpler case, where a wheel is rotating about a fixed hinge. In that
case, the velocities of various particles will be like this (Refer Slide Time: 03:59). There
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are three types of plane motions as you can see here; one is where all the particles on the
rigid body they have exactly same velocity, same motion, identical situation. In another
case, it is a more general situation; it is neither a pure rotation like this nor a pure
translation like this, but a general motion.
(Refer Slide Time: 04:33)

For example, we take our familiar example of a slider crank mechanism, which we have
discussed in our first introductory lecture. Now, we study the three rigid bodies which
are interconnected and they are moving. You can see that this particular rigid body which
you call crank is undergoing a pure rotation about a fixed inch point O. This particular
rigid body is undergoing a pure transitory motion like this (Refer Slide Time: 05:10),
whereas, this one is undergoing both rotation and translation that is a general type of
motion. A machine consists of number of rigid bodies interconnected and each one of
them undergoing different types of motions. Of course, as I mentioned earlier in this
particular example, all the particles of the system they have their displacements,
velocities and accelerations lying in parallel planes, so that the whole motion picture can
be projected on to the projecting plane.



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(Refer Slide Time: 05:47)

The simplest of all these will be the case of translation. A Rigid body where each point
of this rigid body they have exactly same displacement, same velocity, same
acceleration. So, effectively what we can do? Then this is more or less equivalent to a
particle with the same velocity or displacement as the rigid body, but the total mass of
this is same as the mass of this particle. Therefore, cases of pure translation a rigid body
may be considered to be concentrated in a particle with the same displacement
parameters, velocities and accelerations. This is pure translation.
Translation can be of two types: this is a translation which we call, rectilinear translation,
why? Because we find the path of each particle is in the straight line. That means, it was
here; it has gone there. It was here; it has gone there. It was here; it has gone there. It was
here; it has gone there (Refer Slide Time: 05:47).
There is another kind of translational motion is possible, where each particle of a rigid
body undergoes a different kind of trajectory which is not a straight line, but it is same
for all. That means, a particle which was here; it as gone here; particle which was here; it
has gone here; particle which was here; it has gone here and so on (Refer Slide Time:
07:27). Here also at any particular instant of time, the velocity of the particles will be all
identical, but the trajectory of the particles will not be straight lines like this, so this is
called curvilinear translation. The simplest type of motion that is translation can be either
rectilinear or curvilinear, in both these cases, since at every instant all the particles
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undergo identical motion, we can treat the whole body as one single particle like this of
the same total mass as this original one.
(Refer Slide Time: 08:30)

Let us consider an example of space motion. Before, I proceed further with the case of
plane motion, it is also appropriate for us to understand what we call by a space motion?
The simplest example of that will be a spinning top with which we are familiar. We
know that this top undergoes a spinning motion at a very fast rate and the axis of this
undergoes a precessional motion about a vertical axis like this (Refer Slide Time: 09:05).
In this case, we will find that no point on this body or all other points on this rigid body
their velocity vectors, acceleration vectors, etcetera at any particular instant of time do
not lie in single plane or in parallel planes. This is a very common example of space
motion, but in engineering we have cases where a particular system will undergo a
motion where the velocities, displacements and acceleration of various points do not lie
in the same or parallel planes.




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(Refer Slide Time: 10:00)

Let us come back to our study of plane motion; we study plane motion primarily because
a very large number of engineering systems undergo parallel motion problems. As for
example, we have shown our slider crank mechanism, we take a simple crank follower
mechanism, here again we will see that all particles which are describing this system
undergo motion where the velocities, displacements and acceleration lie in parallel
planes. We can have motion of a gear or a wheel or a transmitting power between two
shafts, again you will find that all the points undergo motion where the motion
parameters lie in parallel planes. There are very large numbers of engineering problems
or systems which involve plane motion and so, let us first start studying the problem of
systems involving plane motions. Before we start studying the plane motion problems of
a rigid body, let us see that how the description of the rigid body has to be done or rather
how we describe the motion of a rigid body in plane motion?





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(Refer Slide Time: 11:11)

The next thing we will study in little depth is Description of Motion. What I mean to say
that one object is here and after sometime I find it has gone here, this is the start and this
is the end or a finish (Refer Slide Time:11:29). How to describe the motion of this rigid
body? In case of particle there is no problem, ((in case)) motion of a particle it was start
here and end here. I can simply say that the displacement of this motion is this much and
denoted by vector, but in this case, it constitutes a large number of particles and each
particle may have different displacement vector. For example, suppose if you take this
point, its displacement is this much (Refer Slide Time: 12:01). If you take this point, its
displacement is this much, so they are different, this is called s
1
and this is s
2
, then we
know s
1
is not equal to s
2
. So, how to describe the motion of such rigid bodies that is
what is our problem? To do it, let us see that how we describe the motion of a rigid body.






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(Refer Slide Time: 12:43)

First let us take this rigid body, I am drawing a simpler body for better understanding and
the same body goes there. We can generally show that we split this motion into two
parts, how? First we take any point here, let us say for example, consider these as the
reference points; it can be any point on this, this point P
i
initially has gone to this point as
P
f
, so this point displacement is this and if we just subject this body to a pure rectilinear
translation then the position of this body would have been like this (Refer Slide Time:
13:37). So, this is a pure rectilinear translation of the body from the initial and
intermediate position and then keeping this reference point whatever I have selected, we
give a rotation to the body from this position to this position, so this is the translational
part and this is the rotational part. So, any motion of a rigid body can be split into these
two pure motions; pure translation and pure rotation with one point fixed and that will
provide us this general motion of a rigid body.





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(Refer Slide Time: 14:54)

Now, we will find out the effect of a force on a rigid body, because ultimately, as you
know that our objective is to find out the effect of forces on a machine or the kind of
forces that are generated by motions of machines and machines parts and such things.
Therefore, let us start the effect of force or a system of force on a rigid body, which is in
plane motion. We again take our rigid body and say this is being acted upon by a large
number of forces, which can be all ultimately combined to make this resultant of all the
external forces F, which is acting like this (Refer Slide Time:15:10).
Now, this rigid body is acted upon by this force. Our objective is to find out that what is
the motion of this? We will not go back to the basic fundamentals because that is the
subject which we have studied in the dynamics or this kind of topics or problems we
have solved in dynamics. We will not go into the basic fundamentals of that but certain
important points we will recapture it a little bit for going further.
Now, you have seen in basic dynamics courses that motion of a rigid body due to a force
can be found out in the following manner. First what we do? We identify a particular
point, which you consider as the center of mass. What is the center of mass? The
definition of a center of mass of a rigid body, I will show you here this is C. If you take
any particle whose mass is a delta m
i
and which is located at a position rho
i
, then if you
sum up this product for all the particles; it will result in zero, if C is the center of mass,
this we have to keep in mind, that is the definition of center of mass or if you consider
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that if you take the moment of all the masses about any other point, it can be considered
to be the moment of all the particles about an axis passing through this plus the whole
mass considered to be concentrated here and its moment about that. We will look into it
later.
(Refer Slide Time: 17:42)

Now will go back to our problem, this force which is acting at a distance e from the
center of mass, what we can do now is, we can apply an equal and opposite force minus
F and plus F here, then this becomes equivalent to a force passing through this and a
moment acting on this, which is given by F into e. Now, you will notice that I have not
put a moment vector here because in plane motion all angular activities like whether
angular velocity, angular displacement, moment they can be represented by algebraic
quantities. We may consider that if it is in the clockwise direction, they are positive. If
they are in the anticlockwise direction, they are negative. So, they are summing up or
subtraction becomes or it is just like an algebraic quantity, whereas force, its directions
are different and so they need to be represented by vector.
If the total mass of this be m and second moment of the mass, that is moment of inertia
of the body about an axis passing through the center of mass is, moment of inertia about
the center of mass I
c
. This force will cause an acceleration in this direction of the center
of mass which is a and it will cause an angular acceleration alpha due to this and their
quantitative dependence will be equal to F divided by mass and this will be M divided by
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I
c
. Therefore, any set of forces on a rigid body in plane motion can be considered to be
the effect of a single resultant force on that rigid body, which in turn is equivalent to that
force acting through the center of mass of that body and a moment acting about the
center of mass which is given by the force and the product of the force and the
eccentricity of the force which causes the motion of the rigid body in this following form
that the center of mass is accelerated, which is given by this and the body gets an angular
acceleration alpha, which is given by this, about the axis passing through this (Refer
Slide Time: 20:09). So that means what we are doing? That this rigid body motion is
being described as the acceleration of this center of mass and angular acceleration about
an axis passing through the center of mass which we call C.
(Refer Slide Time: 20:44)

As you know from your basic courses in dynamics that a rigid body, if a particular point
is the center of mass, which plays a central role, its velocity (or) its acceleration and
acceleration angular acceleration about an axis passing through is given that completely
describes the motion of this rigid body, if the initial conditions are known, we can find
out the initial or final values. For example, if we say that initially its velocity was V
0
and
its angular velocity was omega
0,
then at any instant of time the velocity after a time dt, V
will be V
0
plus a delta t and omega will be omega
0
plus alpha into delta t. Therefore, we
can proceed ahead and we can get the velocity at any instant of time, if we know the
initial condition because of this linear and angular accelerations. This brings to the end of
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the case of a single rigid body under the action of a set of forces and what accelerations
are developed. We find out like this.
(Refer Slide Time: 23:05)

Let us find out, how we consider this particular aspect that we have just discussed in
studying the forces which are developed in various machines. As I mentioned in the
introductory lecture that a body when is in motion that means when it is accelerated, then
the system is acted upon by certain dynamics forces which are because of the inertia
forces of the system. To explain this little bit more, let us consider this case.
The simplest case will be again our slider crank and then we will go to more complicated
examples. This is a very useful mechanism that is why we are using it so many problems
the same thing. Now, I tell that this is a pure rotating body about its fixed hinge O and let
its moment of inertia about this fixed hinge be I
o
.
Let us consider this is a pure reciprocating or rectilinear motion body, so I have to give
only mass of this. If the length of this rod is l, radius of this crank is R and at any
particular instant which is shown here like this is theta and say it is being rotated at a
constant angle of velocity omega (Refer Slide Time: 24:16). If you have to design this
particular rod, say our interest is that how do you do it? Now, first what we will do is that
as follows. Let this particular rod, if it is mass less for simplicity, then it will be
subjected to either pure tension or pure compression, which means that resultant force on
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this will be zero, because it is a mass less system it cannot sustain any resultant force or
moment.
When this is weakened from kinematics, you have already found out that the acceleration
of this body at this instant will be a. If that be the situation, then you know that this
particular point that means this rigid body must be subjected to a force in this direction,
which is mass into acceleration. That means, this body its acceleration is a so resultant
force on the body must be F, which is simply mass into acceleration, but we know that
this body is acted upon by these forces. One is, a force from the support and either from
this side, in this particular case it will be from this side, let me correct my drawing. So, it
can be from this and a force which must be acting along this because this rod cannot
sustain any force here which is not in the line of this (Refer Slide Time: 27:00) so, if this
angle is theta or phi, then this angle as to be also phi.
(Refer Slide Time: 27:13)

How much will be this force? That we can find out from this equation P cosine phi must
be equal to mass into acceleration or this force P will be mass into acceleration by cosine
phi. We can find out this reaction because there is no movement along this direction
which is perpendicular to the slide or surfaces, so total force in this direction will be
zero. You can then write that R minus P sine phi is 0. That is, R will be equal to simply P
sine phi. Therefore, we know that this connecting rod which is mass less in this example
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has a force P in this direction. So, obviously here also it must be acted upon by an equal
and opposite force P. So, that this system or this rod is under no resultant force.
This is a particular example and we solved the loading on the connecting rod as just a
simple case of example, but I think this is just from the basic fundamentals we try to
develop the procedure by which you could find out the force, but while dealing
mechanisms with many members more complex motion, a better more generalized
approach of analyzing forces on various members, at various contacts, at various hinges
it will be necessary to develop that methodology and we will next try to develop that
methodology. Before taking up the problem of general mechanism and the problem of
finding out the various forces and moments at various surfaces then the contact forces at
the hinges and so on. We will see that it will be convenient to consider the dynamic
problem after you convert the dynamic problem into a ((strategically)) problem, what I
mean to say is this. Let any system under general motion, if it is in dynamic equilibrium
it means, that it is running under the influence of some forces, then each and every
member of the mechanism will be subjected to some forces which are acting on it and
which are producing the resulting motion parameters like accelerations; both angular and
linear.
(Refer Slide Time: 30:40)

If I say that is a system or a mechanism or a machine is under static equilibrium under
the influence of certain forces. What does that mean? Say for example, if you consider
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this case, a four link mechanism with the ground link fixed, I state that under the action
of this moment, this force and this force, this system is in static equilibrium, if I tell that.
It means that no member will have any motion when this moments and forces are acting.
If the whole system is motion less; it implies that each and every member of this
mechanism or system is in equilibrium that means this link, which I think in your
terminology you have termed it as link four. So, fourth link which is subjected to this
external force we call it say F
4
and there will be two hinge forces and reaction forces
where one reaction force is from here and another reaction force; we call it R
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and this I
call R
34
. Then the combine action of this reaction force, this reaction force and the
externally applied force ease to keep the system under static equilibrium (Refer Slide
Time: 32:26).
Similarly, this particular link which you call the connect coupler or it is under the
influence of this force, which you call F
3
and the reaction force is like this it will remain
under static equilibrium, that is each and every member of this mechanism will be in
static equilibrium under the action of the external forces acting on that particular
member. So, this principle we can use in solving the dynamics problem.
In the next lecture, I will explain the technique of converting a dynamics problem into a
static equilibrium problem and illustrate it with help of an example.
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