Chapter XXIII: Part II: Theory of Machines (3)

These combinations of pieces are known individually as _kinematic
pairs of elements_, or briefly _kinematic pairs_. The three pairs
mentioned above have each the peculiarity that contact between the two
pieces forming the pair is distributed over a surface. Kinematic pairs
which have surface contact are classified as _lower pairs_. Kinematic
pairs in which contact takes place along a line only are classified as
_higher pairs_. A pair of spur wheels in gear is an example of a
higher pair, because the wheels have contact between their teeth along
lines only.

A _kinematic link_ of the simplest form is made by joining up the
halves of two kinematic pairs by means of a rigid link. Thus if A1B1
represent a turning pair, and A2B2 a second turning pair, the rigid
link formed by joining B1 to B2 is a kinematic link. Four links of
this kind are shown in fig. 120 joined up to form a _closed kinematic
chain_.

In order that a kinematic chain may be made the basis of a mechanism,
every point in any link of it must be completely constrained with
regard to every other link. Thus in fig. 120 the motion of a point a
in the link A1A2 is completely constrained with regard to the link
B1B4 by the turning pair A1B1, and it can be proved that the motion of
a relatively to the non-adjacent link A3A4 is completely constrained,
and therefore the four-bar chain, as it is called, can be and is used
as the basis of many mechanisms. Another way of considering the
question of constraint is to imagine any one link of the chain fixed;
then, however the chain be moved, the path of a point, as a, will
always remain the same. In a five-bar chain, if a is a point in a link
non-adjacent to a fixed link, its path is indeterminate. Still another
way of stating the matter is to say that, if any one link in the chain
be fixed, any point in the chain must have only one degree of freedom.
In a five-bar chain a point, as a, in a link non-adjacent to the fixed
link has two degrees of freedom and the chain cannot therefore be used
for a mechanism. These principles may be applied to examine any
possible combination of links forming a kinematic chain in order to
test its suitability for use as a mechanism. Compound chains are
formed by the superposition of two or more simple chains, and in these
more complex chains links will be found carrying three, or even more,
halves of kinematic pairs. The Joy valve gear mechanism is a good
example of a compound kinematic chain.

A chain built up of three turning pairs and one sliding pair, and
known as the _slider crank chain_, is shown in fig. 121. It will be
seen that the piece A1 can only slide relatively to the piece B1, and
these two pieces therefore form the sliding pair. The piece A1 carries
the pin B4, which is one half of the turning pair A4 B4. The piece A1
together with the pin B4 therefore form a kinematic link A1B4. The
other links of the chain are, B1A2, B2B3, A3A4. In order to convert a
chain into a mechanism it is necessary to fix one link in it. Any one
of the links may be fixed. It follows therefore that there are as many
possible mechanisms as there are links in the chain. For example,
there is a well-known mechanism corresponding to the fixing of three
of the four links of the slider crank chain (fig. 121). If the link d
is fixed the chain at once becomes the mechanism of the ordinary steam
engine; if the link e is fixed the mechanism obtained is that of the
oscillating cylinder steam engine; if the link c is fixed the
mechanism becomes either the Whitworth quick-return motion or the
slot-bar motion, depending upon the proportion between the lengths of
the links c and e. These different mechanisms are called _inversions_
of the slider crank chain. What was the fixed framework of the
mechanism in one case becomes a moving link in an inversion.

The Reuleaux system, therefore, consists essentially of the analysis
of every mechanism into a kinematic chain, and since each link of the
chain may be the fixed frame of a mechanism quite diverse mechanisms
are found to be merely inversions of the same kinematic chain. Franz
Reuleaux's _Kinematics of Machinery_, translated by Sir A. B. W.
Kennedy (London, 1876), is the book in which the system is set forth
in all its completeness. In _Mechanics of Machinery_, by Sir A. B. W.
Kennedy (London, 1886), the system was used for the first time in an
English textbook, and now it has found its way into most modern
textbooks relating to the subject of mechanism.

§ 81.* _Centrodes, Instantaneous Centres, Velocity Image, Velocity
Diagram._--Problems concerning the relative motion of the several
parts of a kinematic chain may be considered in two ways, in addition
to the way hitherto used in this article and based on the principle of
§ 34. The first is by the method of instantaneous centres, already
exemplified in § 63, and rolling centroids, developed by Reuleaux in
connexion with his method of analysis. The second is by means of
Professor R. H. Smith's method already referred to in § 23.

_Method 1._--By reference to § 30 it will be seen that the motion of a
cylinder rolling on a fixed cylinder is one of rotation about an
instantaneous axis T, and that the velocity both as regards direction
and magnitude is the same as if the rolling piece B were for the
instant turning about a fixed axis coincident with the instantaneous
axis. If the rolling cylinder B and its path A now be assumed to
receive a common plane motion, what was before the velocity of the
point P becomes the velocity of P relatively to the cylinder A, since
the motion of B relatively to A still takes place about the
instantaneous axis T. If B stops rolling, then the two cylinders
continue to move as though they were parts of a rigid body. Notice
that the shape of either rolling curve (fig. 91 or 92) may be found by
considering each fixed in turn and then tracing out the locus of the
instantaneous axis. These rolling cylinders are sometimes called
axodes, and a section of an axode in a plane parallel to the plane of
motion is called a centrode. The axode is hence the locus of the
instantaneous axis, whilst the centrode is the locus of the
instantaneous centre in any plane parallel to the plane of motion.
There is no restriction on the shape of these rolling axodes; they may
have any shape consistent with rolling (that is, no slipping is
permitted), and the relative velocity of a point P is still found by
considering it with regard to the instantaneous centre.

Reuleaux has shown that the relative motion of any pair of
non-adjacent links of a kinematic chain is determined by the rolling
together of two ideal cylindrical surfaces (cylindrical being used
here in the general sense), each of which may be assumed to be formed
by the extension of the material of the link to which it corresponds.
These surfaces have contact at the instantaneous axis, which is now
called the instantaneous axis of the two links concerned. To find the
form of these surfaces corresponding to a particular pair of
non-adjacent links, consider each link of the pair fixed in turn, then
the locus of the instantaneous axis is the axode corresponding to the
fixed link, or, considering a plane of motion only, the locus of the
instantaneous centre is the centrode corresponding to the fixed link.

To find the instantaneous centre for a particular link corresponding
to any given configuration of the kinematic chain, it is only
necessary to know the direction of motion of any two points in the
link, since lines through these points respectively at right angles to
their directions of motion intersect in the instantaneous centre.

To illustrate this principle, consider the four-bar chain shown in
fig. 122 made up of the four links, a, b, c, d. Let a be the fixed
link, and consider the link c. Its extremities are moving respectively
in directions at right angles to the links b and d; hence produce the
links b and d to meet in the point O_(ac). This point is the
instantaneous centre of the motion of the link c relatively to the
fixed link a, a fact indicated by the suffix ac placed after the
letter O. The process being repeated for different values of the angle
[theta] the curve through the several points Oac is the centroid which
may be imagined as formed by an extension of the material of the link
a. To find the corresponding centroid for the link c, fix c and repeat
the process. Again, imagine d fixed, then the instantaneous centre
O_(bd) of b with regard to d is found by producing the links c and a
to intersect in O_(bd), and the shapes of the centroids belonging
respectively to the links b and d can be found as before. The axis
about which a pair of adjacent links turn is a permanent axis, and is
of course the axis of the pin which forms the point. Adding the
centres corresponding to these several axes to the figure, it will be
seen that there are six centres in connexion with the four-bar chain
of which four are permanent and two are instantaneous or virtual
centres; and, further, that whatever be the configuration of the chain
these centres group themselves into three sets of three, each set
lying on a straight line. This peculiarity is not an accident or a
special property of the four-bar chain, but is an illustration of a
general law regarding the subject discovered by Aronhold and Sir A. B.
W. Kennedy independently, which may be thus stated: If any three
bodies, a, b, c, have plane motion their three virtual centres,
O_(ab), O_(bc), O_(ac), are three points on one straight line. A proof
of this will be found in _The Mechanics of Machinery_ quoted above.
Having obtained the set of instantaneous centres for a chain, suppose
a is the fixed link of the chain and c any other link; then O_(ac) is
the instantaneous centre of the two links and may be considered for
the instant as the trace of an axis fixed to an extension of the link
a about which c is turning, and thus problems of instantaneous
velocity concerning the link c are solved as though the link c were
merely rotating for the instant about a fixed axis coincident with the
instantaneous axis.

_Method 2._--The second method is based upon the vector representation
of velocity, and may be illustrated by applying it to the four-bar
chain. Let AD (fig. 123) be the fixed link. Consider the link BC, and
let it be required to find the velocity of the point B having given
the velocity of the point C. The principle upon which the solution is
based is that the only motion which B can have relatively to an axis
through C fixed to the link CD is one of turning about C. Choose any
pole O (fig. 124). From this pole set out Oc to represent the velocity
of the point C. The direction of this must be at right angles to the
line CD, because this is the only direction possible to the point C.
If the link BC moves without turning, Oc will also represent the
velocity of the point B; but, if the link is turning, B can only move
about the axis C, and its direction of motion is therefore at right
angles to the line CB. Hence set out the possible direction of B´s
motion in the velocity diagram, namely cb1, at right angles to CB. But
the point B must also move at right angles to AB in the case under
consideration. Hence draw a line through O in the velocity diagram at
right angles to AB to cut cb1 in b. Then Ob is the velocity of the
point b in magnitude and direction, and cb is the tangential velocity
of B relatively to C. Moreover, whatever be the actual magnitudes of
the velocities, the instantaneous velocity ratio of the points C and B
is given by the ratio Oc/Ob.

A most important property of the diagram (figs. 123 and 124) is the
following: If points X and x are taken dividing the link BC and the
tangential velocity cb, so that cx:xb = CX:XB, then Ox represents the
velocity of the point X in magnitude and direction. The line cb has
been called the _velocity image_ of the rod, since it may be looked
upon as a scale drawing of the rod turned through 90° from the actual
rod. Or, put in another way, if the link CB is drawn to scale on the
new length cb in the velocity diagram (fig. 124), then a vector drawn
from O to any point on the new drawing of the rod will represent the
velocity of that point of the actual rod in magnitude and direction.
It will be understood that there is a new velocity diagram for every
new configuration of the mechanism, and that in each new diagram the
image of the rod will be different in scale. Following the method
indicated above for a kinematic chain in general, there will be
obtained a velocity diagram similar to that of fig. 124 for each
configuration of the mechanism, a diagram in which the velocity of the
several points in the chain utilized for drawing the diagram will
appear to the same scale, all radiating from the pole O. The lines
joining the ends of these several velocities are the several
tangential velocities, each being the velocity image of a link in the
chain. These several images are not to the same scale, so that
although the images may be considered to form collectively an image of
the chain itself, the several members of this chain-image are to
different scales in any one velocity diagram, and thus the chain-image
is distorted from the actual proportions of the mechanism which it
represents.

§ 82.* _Acceleration Diagram. Acceleration Image._--Although it is
possible to obtain the acceleration of points in a kinematic chain
with one link fixed by methods which utilize the instantaneous centres
of the chain, the vector method more readily lends itself to this
purpose. It should be understood that the instantaneous centre
considered in the preceding paragraphs is available only for
estimating relative velocities; it cannot be used in a similar manner
for questions regarding acceleration. That is to say, although the
instantaneous centre is a centre of no velocity for the instant, it is
not a centre of no acceleration, and in fact the centre of no
acceleration is in general a quite different point. The general
principle on which the method of drawing an acceleration diagram
depends is that if a link CB (fig. 125) have plane motion and the
acceleration of any point C be given in magnitude and direction, the
acceleration of any other point B is the vector sum of the
acceleration of C, the radial acceleration of B about C and the
tangential acceleration of B about C. Let A be any origin, and let Ac
represent the acceleration of the point C, ct the radial acceleration
of B about C which must be in a direction parallel to BC, and tb the
tangential acceleration of B about C, which must of course be at right
angles to ct; then the vector sum of these three magnitudes is Ab, and
this vector represents the acceleration of the point B. The directions
of the radial and tangential accelerations of the point B are always
known when the position of the link is assigned, since these are to be
drawn respectively parallel to and at right angles to the link itself.
The magnitude of the radial acceleration is given by the expression
v²/BC, v being the velocity of the point B about the point C. This
velocity can always be found from the velocity diagram of the chain of
which the link forms a part. If dw/dt is the angular acceleration of
the link, dw/dt × CB is the tangential acceleration of the point B
about the point C. Generally this tangential acceleration is unknown
in magnitude, and it becomes part of the problem to find it. An
important property of the diagram is that if points X and x are taken
dividing the link CB and the whole acceleration of B about C, namely,
cb in the same ratio, then Ax represents the acceleration of the point
X in magnitude and direction; cb is called the acceleration image of
the rod. In applying this principle to the drawing of an acceleration
diagram for a mechanism, the velocity diagram of the mechanism must be
first drawn in order to afford the means of calculating the several
radial accelerations of the links. Then assuming that the acceleration
of one point of a particular link of the mechanism is known together
with the corresponding configuration of the mechanism, the two vectors
Ac and ct can be drawn. The direction of tb, the third vector in the
diagram, is also known, so that the problem is reduced to the
condition that b is somewhere on the line tb. Then other conditions
consequent upon the fact that the link forms part of a kinematic chain
operate to enable b to be fixed. These methods are set forth and
exemplified in _Graphics_, by R. H. Smith (London, 1889). Examples,
completely worked out, of velocity and acceleration diagrams for the
slider crank chain, the four-bar chain, and the mechanism of the Joy
valve gear will be found in ch. ix. of _Valves and Valve Gear
Mechanism_, by W. E. Dalby (London, 1906).