- Introduction
- Centros
- The Crank and Slider Mechanism
- The Pantograph
- Straight-Line Mechanisms
- Curious Mechanisms
- Helixes and Screws
- Belt Drives
- Motion
- Centripetal Acceleration Graphically
- Making Models of Linkages
- References

At least until the 1970's, an undergraduate course in "the elements of mechanism" was part of the core engineering curriculum, generally appearing in the latter half of the second year. This course was also called "kinematics." It probably then became a part of the mechanical engineering curriculum, no longer taken by all branches, and then was eliminated as graphical methods were abandoned. This is quite a shame, as we shall see, because of the interesting fundamental material that was presented in this course, such as gears and gearing, threads, and many common mechanical definitions. Now mechanisms appear piecemeal and in very attenuated form in other courses, such as mechanics or machine design, to reappear in graduate studies as a theoretical course of dubious real-world applicability. The elementary parts are scorned by the modern academic "engineers" who now teach undergraduates, and haven't seen too many machines or mechanisms themselves.

The prominence of graphical methods in the mechanisms course was one thing that led to its demise. Graphical methods are not now popular for engineering calculations, even where they might be perfectly adequate. Instead, reliance is placed on computer calculations, which are invariably accurate if the programming is correct, and have the added advantage of giving results to extreme precision, if not to extreme accuracy. This is fine for manufacturing, but not for understanding, ingenuity and conceptual thinking. Graphical methods do not only present results clearly and evidently, they also train the skills of spatial thinking and reasoning, and this is perhaps their greatest advantage. The association of movement of the hands with intellectual activity is of fundamental value, much more efficient than simply inertly reading a text. Computers just produce numbers, not insight, as has often been said. Graphical methods are also fun.

In this article, I shall look at some interesting topics from an introduction to mechanism course, by no means giving a balanced or complete account. Information on gears can be found in Trochoids, Etc.. Planetary (epicyclic) gear trains are explained in Planetary Gears. This is a notoriously confusing subject that requires some crafty methods for the solution of problems. Machines, a subject closely related to mechanism, are discussed in Machines.

A *machine* is a device for applying power, while a *mechanism* is a mechanical device considered purely with respect to its motions. Or, a machine modifies *energy*, while a mechanism modifies *motion*. Machines may, and usually do, consist of mechanisms, with a source of power added. The science of mechanism may also be called *kinematics*. A *structure* is a mechanism in which motion is precluded. A *lower pair* consists of two elements in surface contact, where one body surrounds the other, as a piston in a cylinder or a shaft in a journal. A *higher pair* consists of two elements in line or point contact, as in ball bearings or a wheel on a rail. To *invert a pair* means to exchange fixedness between the two members. In a lower pair, inversion has no effect on the relative or absolute motion. In higher pairs, inversion has no effect on the relative motion, but the absolute motion is different. For example, if a wheel rolls on a fixed rail, a point on the circumference traces a cycloid. If the wheel remains still, and the rail rolls on it, the point of contact (on the rail) traces an involute. In an *incomplete pair*, contact is maintained by external forces. A wheel is an incomplete pair. In general, a *driver* moves a *follower*. A body with rotational movement is often said to *revolve* about an axis outside the body, but to *rotate* about an axis passing through the body. For example, the earth revolves about the sun in its orbit, but rotates about its axis.

The basic four-bar linkage is shown at the left. It consists of cranks 2 and 4 rotating on shafts fixed in the frame 1, and connected by link 3. The mechanism itself is shown in configuration space at the left in the diagram. On the right is a *graph* of the mechanism, in which the links of the mechanism are represented by the vertices, and the joints by the lines. The problem of determining all possible graphs for a given number of links is an interesting problem in topology, which has received a good deal of attention. This exercise is called *type synthesis*. Although interesting and difficult, it is probably not of much practical importance, though it has its enthusiasts. Such matters were not presented in the undergraduate mechanisms course.

A planar linkage with L links and J joints has F = 3L - 2J - 3 *degrees of freedom*. The number of degrees of freedom is the number of generalized coordinates that must be specified to determine the configuration of the linkage. For the four-bar linkage, L = 4 and J = 4, so F = 12 - 8 - 3 = 1. We will consider here only linkages with one degree of freedom, and the four-bar linkage will be the principal example. One very useful form of the four-bar linkage is the crank and slider mechanism, in which crank 4 is of infinite length, so that the joint has a reciprocating motion.

The study of mechanisms can be extended beyond its classical limits to include other kinds of "links," such as pneumatic, hydraulic, electrical or electronic. This is a very valuable and interesting extension, but will not be followed here. Mechanism also is divided into *analysis*, the description of an existing mechanism, and *synthesis*, a much harder matter, in which a mechanism is designed to perform a certain task. Analysis must be thoroughly understood before tackling synthesis.

Mechanism includes not only rigid links connected by rotating joints, but also cams and gears where the motion is transmitted by sliding surfaces, flexible cords, belts and chains, and logical elements, such as ratchets, trips, detents, escapements and interlocking mechanisms. It is also convenient to include the effects of static friction in mechanism. The consideration of forces *as they produce motion* is part of dynamics, and dynamics borders directly on kinematics or mechanism.

The term "centro" is certainly one that the modern US engineering graduate, and most of his or her teachers, will not recognize. It does not even appear in the *Cambridge Dictionary of Science and Technology*. It arises in the solution of the problem of finding the velocity of one member of a mechanism when the velocity of another is given. We will consider a mechanism as an arrangement of connected members in a plane, such that the motion of one member determines the motion of the others. Our example will be a very basic mechanism, the four-bar linkage. It will be arranged so that bars 2 and 4 rotate about fixed axes in bar 1, while connected by bar 3. Each bar will be imagined as part of a plane member of indefinite size, and we will not worry about any mechanical interference that this causes. Imagine the four plates stacked up and connected at the fixed points 12 and 14 (where the two numbers indicate the members connected at that point), and the moving points 23 and 34. It should be clear that each of these four points have the same velocities in both of the members they connect, as guaranteed by the axes or pins.

Now consider the cranks 2 and 4. Assume they are disconnected from member 3 and rotate independently about points 12 and 14. The velocity of any point in member 2 is proportional to its distance from 12, and the velocity of any point in member 4 is proportional to its distance from 14. Now consider a line that passes through 12 and 14. There will be some point on this line, depending on the ratio of angular velocities of 2 and 3, that has the same velocity in members 2 and 4.

Now draw a line passing through points 23 and 34, and imagine members 2 and 4 as rotating about these points rather than about 12 and 14. There must be some point on this line that has the same velocity in member 2 and member 4. If we now connect the mechanism, so that the movement of the two cranks is simultaneously constrained by members 1 and 3, there will be one point with the same velocity in members 2 and 4, and it must lie at the intersection of the two lines 12,14 and 23,34. This point will be called 24. This point is not permanently fixed in either body, and is not determined by a pin or shaft. Nevertheless, at any instant of time, it is there, and there is only one such point.

Note that we can find this point by drawing the two lines through points where we can "cancel" a common number. That is, 12 and 14 → 24, and 23 and 34 → 24. The reader should now be able to see that this rule also determines a point 13 that has the same velocity in members 1 and 3. "2" and "4" are canceled to find 13. This is called *Kennedy's Theorem*.

These points of the same velocity in pairs of members are called *centros*. In this case, there are six centros. Centros 12 and 13 are fixed in the supposedly fixed member 1. Centros 23 and 34 are at the ends of the connecting member 3. Centros 24 and 13 are found by Kennedy's Theorem, and their locations depend on the configuration of the mechanism. As might be expected, 24 and 13 are the useful centros. Now we will see how centro 24 can be used to determine the relative velocities of cranks 2 and 4, which, of course, rotate about their axes 12 and 14.

First, the mechanism 1234 is drawn in the desired position, as in the diagram. Then, it is easy to locate the centros, which is really the most essential role of the drawing. The reader is invited to determine them with analytical geometry to see which method is the easier and more apparent. Now, the velocity of point 23 is laid off perpendicular to crank 2. This determines the speed of rotation of member 2, and so the velocity of centro 24, which can be determined graphically by laying off similar triangles. V_{23} is laid off at point a. A line from 12 through the tip of V_{23} intersects a vertical line from 24 at c, determining V_{24}. A line from 14 to point c intersects a vertical line from point b, giving V_{34}, which is then transferred to point 34, normal to crank 4.

Since member 1 is not in motion, centro 13 must be fixed in space. It is, therefore, the instantaneous centre of rotation of the connecting link 3. From the velocity at either end of member 3, the angular velocity of rotation can be found. Of course, normals to the velocities at the ends of the link pass through the centro 13, as can be seen in the diagram.

The velocity analysis of linkages depends on the fact that the members are rigid bodies, so that the distance between any two points is constant. This means that the relative velocity of any two points is normal to the line joining the two points. The method of centros handles this very simply, by considering only rigid rotations about certain axes. Another method of analysis uses a vector diagram of velocities. Each velocity vector in the diagram is perpendicular to the member to which it refers. Absolute velocities are drawn from a single pole Q.

Consider the four-bar linkage shown at the left. Suppose V_{23} is known, and V_{34} is to be found. From point Q, draw V_{23}. The relative velocity of 34 with respect to 23 must be normal to the link 3, which is in the direction of line a-b. V_{34} must be normal to the crank 4, so a line is drawn in this direction in the vector diagram. These two lines intersect at point b, and so V_{34} is determined. This is as easy as the method of centros, but uses a graphical vector diagram.

The familiar and important crank-and-slider mechanism is a special case of the four-bar linkage where the radius of one crank is infinite, which means that the joint 34 moves on a straight line, as shown in the figure. Member 2 is the crank, member 3 is the connecting rod, and the slide guiding joint 34 is fixed to the frame, member 1. The path of 34 need not pass through 12, but in most practical mechanisms it does, so that the motion is symmetrical and as simple as possible. It is easy to find the centros 24 and 13 for this mechanism. 24 lies on a vertical line through joint 12, which guarantees that in any case its velocity is parallel to line 12-34, as constrained by the slider. 13 lies on a vertical line through 34. The velocity of 34 can be found either from the motion of centro 24, or from a vector diagram, as shown. V_{23} is drawn from Q. A line parallel to 12-34 gives the direction of V_{34}, and its magnitude is determined by the intersection of this line with a line perpendicular to the connecting rod.

Imagine this diagram for different crank angles θ and see how it gives the velocity of the slider. For θ = 0, 24 is on the line 13-34, so its velocity must be zero. For θ = 90°, 24 is at the end of the crank, so V_{34} = V_{23}.

Let a be the length of the crank, and s the length of the connecting rod. By trigonometry, we find that a sin θ = s sin φ gives the relation between the inclination of the connecting rod φ and the crank angle θ. The position of 34 is then x = a cos θ + √(s^{2} - a^{2}sin^{2}θ). We can find the velocity by differentiating, assuming that θ = ωt, but already we have a complicated expression from which we can get numbers, but little insight.

The ratio a/s is often relatively small in practice. If this ratio is zero, then our expressions give φ = 0, and x = s + a cos θ. The joint 34 then moves with simple harmonic motion. To get an idea of how the actual motion for a finite a/s differs from this, we can expand in powers of a/s. The result is x = s + a^{2}/4l + a cos θ - a^{2}/4l cos 2θ. We notice that the obliquity of the connecting rod introduces a second harmonic into the motion that flattens one loop of the cosine curve and extends the next one, and that the average distance is increased by a small amount over s. This is a good example of what we can do with algebra to make things more understandable.

A variation of the crank and slider has an oscillating slider whose axis is parallel to the connecting rod, but rotates about a fixed pivot. The slider may be a cylinder in which a trunk piston works. This mechanism is often found in toy steam engines, where the motion of the cylinder acts to admit and exhaust steam without any other valve gear.

The *pantograph* (Greek: "all-writer") is a four-bar mechanism used to enlarge or reduce drawings. Its basis is a parallelogram of four links with joints ABCD, as shown in the figure. Link AB is extended to E, which is a fixed point. Link BC may be extended to a point F. The line EF intersects two links at points H and G. F may be anywhere on CD or CD extended, and points H,G will lie on the corresponding links or their extensions. The key to understanding how the pantograph works is the realization that however F may move, the line FE will always pass through H and G. This is certainly not obvious, but a model of the mechanism will demonstrate that it is true. The proposition can be proved by considering similar triangles. If it is true, then it is clear that the motions of F, G and H will be proportional to their distances from the fixed point E. If F moves to F', then H will move to H', for example, and FF' and HH' will be parallel and in the ratio of FE to HE. In fact, this ratio will also be that of EA to EB. Therefore, if F traces out a figure, H will trace out a similar, but reduced, figure. If H traces out a figure, then F will trace out a similar, but enlarged figure.

The current collector of an electric locomotive is called a pantograph, from its construction of slender links that move like a pantograph, although it is not a pantograph at all, but an example of *lazy tongs*. If s is the length of a link in lazy tongs of N sections, then they will extend a distance of x = 2Ns sin θ, where θ is the angle shown in the figure. It is easy to find the velocity ratio dx/dθ, and so the "leverage" of the tongs.

A current collector, or "pantograph," is illustrated in the diagram. Note the spring that extends the collector, and the link at the bottom that causes the two sides to move together. The operating mechanism, that retracts the collector pneumatically, is not shown. Of course, there are many variations, but this is typical. There are two such mechanisms, one on each side, with transverse bracing. The structure is kept light, so that if it is swept away it does not do excessive damage and breaks away cleanly. It must exert a constant pressure about independent of the height of the wire, and must adapt to changes in height quickly without bouncing.

Thomas Newcomen's atmospheric engines used a walking (or rocking) beam with arch heads to guide the piston and pump rods vertically. There was only tension on these connections, so a flexible chain around the arch head was sufficient. The walking beam also converted the downward power stroke of the cylinder to the upward pumping stroke. Watt's earliest engines of 1776, which were also pumping engines, used arch heads to guide the piston rod. These engines not only had separate condensers, so the cylinder could be kept hot, but also the cylinders were bored with Wilkinson's new (1774) boring machine. These improvements made them much more efficient than even Smeaton's improved Newcomen engine.

However, the real need was for a double-acting rotative engine, for which the arch head with flexible band was not suitable for guiding the piston rod on the upward stroke. Watt first considered a linear rack on the piston rod driving gear teeth on the arch head, but this was too heavy and noisy (see Hills, References, for an illustration). An obvious solution is to use linear guides, which became the eventual solution. The available machine tools could not make accurate flat surfaces, so the type of guides later used were ruled out. Other engine makers, such as Trevithick, used circular guides, which could be turned on a lathe, with cylindrical sliders.

To solve this problem, Watt turned to a four-bar linkage, shown in the figure. Point P is guided along the dotted straight line for a stroke of length S by the three links shown. The fourth link is, of course, the frame in which pivots A and D are fixed. Links AB and CD need not be the same length. The point P is at the intersection of link BC with the line HF. The length of link AB should be AB = AF + S^{2}/16AF, and link CD should be of length CD = DH + S^{2}/16DH. Also, FB/HC = FP/HP = BP/CP. Things are simplest when AB = CD, and P is the midpoint of BC. This mechanism was patented in 1782.

Watt applied this linkage so that pivot D was the journal of the walking beam, and pivot C was another point on the walking beam, about halfway to its end. Pivot A was supported on the frame of the engine, beyond or at the end of the walking beam. A pantograph parallelogram was then used with BC as one side, and the other side parallel to the beam, from B to the point E where the connecting rod was attached (which might be just behind the fixed pivot A). A line from the journal of the rocking beam (point D) to point E was constrained to pass through the point P that the Watt mechanism guided in a straight line, from the principle of the pantograph. This straight line was duplicated and enlarged at point E. This is a very ingenious arrangement, and one that casual observers often do not appreciate. On the actual engines, the point P is often not represented, but a nearby joint for connecting a pantograph bar may be confused with it (see Hills, in the References, for a good drawing).

Watt also invented the trunk engine in 1784, where the piston is guided by the cylinder. This principle is used in the reciprocating internal combustion engine, and was also used in oscillating-cylinder engines. After Pickard's patent (see below) expired in 1794 (the normal patent term was 14 years), Watt engines were made with cranks, which after all is simpler than the sun-and-planet gearing.

Boulton and Watt made 496 engines at the Soho Foundry while their patent on the steam engine was effective, 1775-1800 (the extended patent term was due to an Act of Parliament). (There were six steam engines in all in the United States in 1800.) With the expiration of the patent, the way was also open to the development of high-pressure steam engines that did not require a condenser. Watt engines all used a steam pressure less than 15 psi, including those used on Fulton's Hudson River steamboat of 1807. Richard Trevithick (1771-1833) of Cornwall, and Oliver Evans (1755-1819) of Philadelphia, are the most famous pioneer designers of high-pressure engines, and in particular, engines that could move themselves. Both Trevithick and Evans built their first engines around 1801 or 1802. Evan's *Orukter Amphibolos* ("Amphibious Digger") of 1805, which had a single cylinder, half-beam motion, and a pressure of 50 psi, was an example. Evans's engines, greatly modified, were used in the high-pressure steamboats of the western rivers in the United States, and remained very primitive as long as they were used. Trevithick ran the small *Catch Me Who Can* on a circular track in Euston Square in London in 1808. The high price of fodder during the Napoleonic wars stimulated the development of steam locomotives in the coal fields of England and Scotland.

A straight-line motion often used with high-pressure engines was the *half-beam* mechanism, shown in the figure. In order that P be guided on a straight line, the length of link AB should be the mean proportional between the lengths CB and BP. The link CD should be as long as convenient; ideally, C should move in a straight line. Evans used such a mechanism on his engines. It was used on the locomotives of Wylam Colliery built by William Hedley, notably the *Puffing Billy* of 1813, which had vertical cylinders and drove the wheels through cranks. The engines on each side of the locomotive were 90° out of phase, to avoid dead spots, and the large half-beams above the engines had a shape and syncopated motion that was likened to that of a grasshopper's legs, and so the engines were called "grasshoppers." This name spread to the mechanism itself in later years. The mechanism also goes under the name of Scott Russell, the eminent naval architect. It is said to have been invented by William Freemantle in 1803. Some of Hedley's engines had Watt parallel motion with a half-beam instead of the half-beam parallel motion.

The first locomotives of the Stockton and Darlington were four four-wheeled coupled machines like the famous Locomotion No. 1, put into service in September, 1825. These had two cylinders, but one cylinder drove the front axle, while the other drove the rear axle, 90° out of phase or "quartered." Each had its own half-beam parallel motion on top of the boiler. The cylinders were 9-1/2" x 24", pressure 50 psi, wheels 48" diameter. These locomotives pulled coal trains, and the driver generally walked beside them. By 1827 Timothy Hackworth had built the "Royal George" for the S. & D., a six-wheels coupled engine, with 11" x 20" vertical cylinders on the rear of the boiler driving the crank pins directly through a connecting rod, pistons guided by a simple and effective half-beam motion. These vertical cylinders caused the engines to "pitch," or rock up and down, but this was not detrimental at the low speeds at which they were used.

The next year, Stephenson made the *Lancashire Witch*, with inclined 9" x 24" cylinders and slide bars, but in 1829 Foster and Rastrick made the famous *Stourbridge Lion*, the first locomotive in the United States, which had rather prominent half-beams guiding its 8-3/4" x 36" cylinders. The connecting rods ran down from the half-beams. This engine, though weighing only 7 tons or so, was too heavy for the wooden 4' 3" gauge track of the Delaware and Hudson Canal Company, and spent its life as a stationary engine.

In the same year, the Rainhill trials on the Liverpool and Manchester were won by Stephenson's *Rocket*, which had slide-bar guides and inclined cylinders. After this, locomotives had horizontal cylinders and slide-bar guides, except for the occasional oddity, since they had become much faster. The Stockton and Darlington's *Swift* of 1836 had vertical cylinders and Watt straight-line motion driving a jackshaft, from which the wheels were driven. The year 1830, then, is about the epoch when the linkage straight-line motion finally disappeared, at least for locomotives. It lingered on for many years with stationary engines, especially those with walking beams and vertical cylinders. It was thought for many years that horizontal cylinders would wear unevenly, a fear that turned out to be unwarranted. The main defect of the straight-line motions was their ungainly size and weight, especially for fast engines, compared to a compact slider mechanism.

About this time, mathematicians became interested in the problem of straight line motions, and developed many mechanisms of great ingenuity, but of very little practical use. The famous mathematician Chebyshev was very interested in the problem, among many other problems in mechanisms, and one of the straight-line motions he devised is shown in the figure. It is a four-bar mechanism, with crossed links. It is almost simple enough to be a practical mechanism, but no use was available for it. Like the Watt and half-beam mechanisms, it gives an approximate straight line. However, these mechanisms all give lines you will not be able to distinguish from straight over a considerable distance.

The eight-bar linkage shown in the figure on the right moves P in an exact straight line. It was invented in 1864 by Peaucellier, who called it a *compas composé*. Two forms are given; it is quite common for a mechanism to be realized in more than one way. The relationship between the two forms shown is evident. This mechanism was actually used in a blowing engine to air condition the Houses of Parliament in London, since one of the MP's was highly impressed by it, as was Lord Kelvin.

Gears can also be used for straight-line motions. A symmetrical pair of equal gears in mesh can move a crosshead in a straight line by connecting rods from each gear. The hypocycloid formed by gears in the ratio 2:1 is a straight line, the diameter of the larger internal gear. These had some application in small machines, but were mainly curiosities. There are many other straight-line mechanisms; this has been only a selection of the more famous.

This section will mention some mechanisms that are not necessarily curious or surprising, but are famous, ingenious and worth knowing. In the last half-century, some very ingenious and remarkable mechanisms have become obsolete, replaced by electronic devices of much less cost. Ones that come to mind are the mechanical watch, the mechanical calculator, and the typewriter or teletypewriter. The mechanical clock dates from about 1300, but the other mechanisms are modern. All are worth a look as examples of the highest development of the mechanical art. The Curta hand-held mechanical calculator was featured recently in *Scientific American* (see References). Valve gear for steam engines, and the mechanical interlocking of points and signals on railways are two other applications of mechanism, and are rather involved and difficult problems as well. The textile and agricultural industries offer many examples of ingenious mechanism to produce the necessary motions.

When Watt was designing his double-acting rotative engine in 1780, mill owner James Pickard of Birmingham patented an improved rotative atmospheric engine, using a flywheel to maintain the rotation, and a crank to convert reciprocating motion to rotary motion. There was certainly "prior art" here, as cranks had long been used on foot-operated lathes and spinning wheels. Also, there is a suspicion that Pickard had been tipped off about Watt's coming need for the crank. Whatever the case, Watt decided not to fight the patent and devised an excellent substitute he called the "sun-and-planet" mechanism, which no court could mistake for a crank. This mechanism, shown in the figure, is an example of an epicyclic gear train. Let gear A have N teeth, and gear B, N' teeth. Gear B is rigidly connected to the output shaft, and gear A is constrained to move in a circular path around gear B by a circular groove. (An arm could have been used, but then it would have looked like a crank, which was the patented thing.) Gear A is rigidly connected to the connecting rod of the engine. It does not rotate, but only oscillates somewhat back and forth. When gear A has made one complete revolution about the axis (not a *rotation*), how many turns has gear B made?

To solve this problem, we consider the actual motion as the sum of two separate simpler hypothetical motions. First of all, let's hold the arm stationary, and give gear A one turn anticlockwise. Then gear B makes N/N' turns clockwise. Second, suppose the gear train is locked, and we rotate everything one turn clockwise. If we sum these two motions, gear A makes no turns at all (which is what really happens) while gear B makes N/N' + 1 turns. Therefore, the answer is N/N' + 1. If, as in Watt's example, N' = N, the shaft makes two revolutions for each stroke of the engine. His first rotative engine (1788) worked at 25 double strokes per minute, which gave 50 rpm.

The sewing machine also includes interesting mechanisms. For many years, sewing machines were driven by a treadle through a connecting rod and crank. However, the most interesting part is how the stich was formed. A lock stitch requires that one thread be wrapped around the other, something easy to do by hand, but difficult for a machine. Elias Howe (1819-1867) used a shuttle for this purpose, as in a loom. He was driven out of Boston, and his machine smashed by irate tailors, but it had a better reception in England. Isaac Singer (1811-1876) greatly improved the sewing machine by automatically and intermittently advancing the cloth, and by replacing Howe's shuttle with the bobbin. One thread comes from a reel on the top of the machine, passing through a hole in the tip of the reciprocating needle, while the other comes from a small bobbin beneath the presser plate. When the needle descends, it forms a loop of thread below the cloth, which is picked up by a rotating hook that catches it and passes it around the bobbin before releasing it. The stitch is tightened when the needle rises again, and the cloth is moved forward in preparation for the next stitch.

A four-bar linkage with two sliding pairs is shown at the left. Only joints A, fixed, and B, moving, are at a finite distance. The mechanism is called a Scotch Yoke in the U.S, or a Donkey Crosshead in Britain, and gives true simple harmonic motion if the crank AB rotates uniformly. The displacement of the yoke from its central position is x = s cos θ. The yoke can, of course, be supported symmetrically. It works best if the crank is the driver, and the yoke the follower, since this is "following motion" in the same sense as for gears. If it is driven by the yoke, then a flywheel is necessary on shaft A to get the mechanism past the dead centres. There are other forms of four-bar mechanisms with two sliding pairs, for example with C at a finite distance, so the slot in the yoke is curved.

The helix is an interesting three-dimensional curve that plays a significant role in mechanism and machines. If the z-axis is taken as the axis of the helix, then it can be defined in terms of a parameter θ by θ = 2πz/L, x = a cos θ and y = a sin θ, where x,y,z form a right-handed coordinate system. This is a *right-handed* helix. If x = a cos θ and y = -a sin θ the helix is *left-handed*. Right- and left-handed helixes are mirror images of each other. L is called the *lead* of the helix; it is the axial distance for one complete turn.

A *screw thread* is a helical ridge and groove either externally on a cylindrical *screw* or *bolt*, or internally on a cylindrical *nut* that mate with each other by rotation. With right-handed threads, if we are holding a bolt intending to screw it into a nut, then a clockwise rotation of the bolt causes it to advance into the nut. Similarly, if we are holding the nut intending to screw it onto the bolt, a clockwise rotation of the nut causes it to advance. For a left-hand thread, an anticlockwise rotation produces an advance. These are simple matters, but it is good to try to state them clearly. Most threads, especially for fasteners, are right-handed. The nomenclature of screw threads is shown at the left. The pitch is the reciprocal of the number of threads per axial unit length. The major or nominal diameter is the size of hole through with the screw will just pass.

Threads may be produced by cutting, rolling or casting. Cut threads are the most accurate, rolled threads are the strongest, and cast threads the cheapest. External threads may be cut on cylindrical stock with a *die*, a kind of nut with hardened cutting threads and a means of disposing of the cuttings. Internal threads are made with a *tap*, a kind of bolt with hardened cutting threads. As the stock enters the die, or the tap enters the hole, the threads are cut more and more deeply until they have the proper form. Larger threads may also be cut on a lathe, which can cut threads of any form, either right-handed or left-handed, internal or external, but at a higher expense than a die or tap. Henry Maudslay patented a modern screw-cutting lathe in 1797, and this was the effective beginning of the widespread use of screw threads in engineering. Cast threads are commonly used with plastic. Cast iron gears have long been used for slow speeds.

Thread profiles for fasteners are shown at the right. They are derived from the V-thread, with crests and roots rounded off to make manufacturing easier and to avoid sharp corners for better fatigue strength. Thread heights are about 5/8 of the height of a V-thread of the same pitch. The V profile aids firm tightening. Power or motion threads are based on a square thread instead. Their height is generally half the pitch, and the included angle is 10° to 29°. The American National thread, a development of the Sellers thread (1864) has a 60° angle and flat crest and root, while the Whitworth (1841) has a 55° angle and a rounded crest and root. The Whitworth thread, long used in England, is slightly superior to the Sellers thread but is more difficult to manufacture. The Unified thread is a compromise, with a 60° angle but a rounded root. The crest may be rounded or flat. Metric threads resemble the Unified thread closely. A Unified thread is specified as, for example, 1/4-20 UNC, meaning D = 1/4", 20 threads per inch, coarse series. A metric thread is specified as, for example, M8 x 1, meaning major diameter 8 mm, pitch 1 mm. The smallest metric screw is M1.6 x 0.35, and the largest is M100 x 6. The M8 x 1.25 was an older metric standard, almost identical to the 1/4-20 UNC. There are many fewer standard metric sizes than Unified inch sizes. Metric and Unified threads of about the same size will work together, if imperfectly, because the threads have about the same shape.

In inches, an M1.6 x 0.35 screw is 0.063 x 73 threads per inch, very close to what would be an 0-72 UNC, the smallest size in this U.S. series. There is a smaller screw, the 00-90 used in eyeglass frames and similar places, but this does not seem to be a part of the standard screw series. The smallest fine series thread is 0-80 UNF, with D = 0.060" (about 1.16"). Numbered sizes go from 0 to 12, and then the next size is 1/4". The coarse series includes 1-64, 2-56, 3-48, 4-40, 5-40, 6-32, 8-32, 10-24, 12-24 and 1/4-20. The fine series includes 0-80, 1-72, 2-64, 3-56, 4-48, 5-44, 6-40, 8-36, 10-32, 12-28 and 1/4-28. The fine series should generally be used in sizes 0 and 1, the coarse series for larger sizes, except where there is a special reason for using the fine series.

A helix can be considered as an inclined plane wrapped around a circular cylinder. If φ is the slope angle of the inclined plane, then tan φ = L/2πa = L/πD, where D is the diameter of the cylinder. Consider a helix developed as an inclined plane. The inclined plane is repeated every interval of πD for a single helix. Now imagine a second helix started at πD/2 of the same slope; its turns fit in between the turns of the first helix. This is called a *double helix*, and threads cut like this are called *double threads*. Clearly, the helix advances by the same amount per turn, L for a double as a single helix. However, the axial distance between threads, or the *pitch*, is half what it was. Threads may be double, triple or any multiple for which there is enough room between the threads. This gives a strong thread that advances rapidly when turned. My half-gallon orange juice jug has a cap with a nine-tuple thread, and closes in less than half a turn.

Screws are used for fastening, adjusting, and for motion or power. The use of metal screws for fastening is a rather recent development, made possible only by machine tools that produce threads interchangeably, easily and cheaply. This occurred in the 19th century; before that, there were no wood screws or machine screws, and every screw-nut pair had to be individually made. Screws have, however, been known since antiquity and applied mainly to the production of motion, as in the Archimedean screw pump, whose manufacture is described in Vitruvius, in the screw press or in worm gears. Threads had to be tediously filed out, whether in wood or metal, until the appearance of machine tools. The making of an internal thread was much more difficult than making an external thread.

The use of threads for adjusting depends on the possibility of controlling small motions by rotation of the nut or screw, and on the holding of a setting through friction. An example of very accurate adjustment screws were the screws made by Rowland for the ruling of diffraction gratings. Clamps and screws were used for the accurate setting of theodolites and similar instruments.

Motional applications of screws are quite common. The screw jack is an example, in which friction will safely hold an elevated heavy object, and by means of which a large force can be exerted with minimal effort. A good example is the lead screw of a lathe, which drives the carriage holding the cutting tool. If the lead screw is geared to the spindle, then an external thread can easily be cut, making several passes until the thread shape is complete. An internal thread is bored in an analogous way.

Fasteners may occur as bolts, passed through a hole in both members to be joined and secured by a nut; studs, screwed into a tapped hole in one member and secured by a nut; and cap screws, passed through a hole in the member to be attached and screwed into a tapped hole. Machine screws are simply small fasteners that can be used as bolts or cap screws. Bolts and screws of all sizes may have hexagonal or square heads, with hexagonal or square nuts, that can be rotated with a "wrench" or "spanner." Nuts may be regular, heavy or jam nuts. A jam nut is thinner than a regular nut, and is used beneath a regular or heavy nut to lock the combination. The upper nut must be tightened enough that the two nuts bear on the threads in opposite senses, which is easier to do if the lower nut is thinner.

Machine screws may have a large variety of forms of heads, slotted or recessed, that are turned with a driver. The cross head, one form of which is the Phillips, was developed to keep the driver in the slot better in machine driving. Socket heads may be hexagonal or splined, and are widely used on headless set screws as well as on machine screws. A bent hexagonal rod, an "Allen wrench," is used as a key with hexagonal sockets. The head shape itself may be round, flat (fitting in a countersunk hole), fillister (cylindrical, for extra strength), pan (flat top and bottom), oval (combining round and flat), truss (thin, large diameter), or other. A *binding head* is recessed at the centre so that the head is strained when fully driven and acts like a lock nut.

Self-tapping screws are driven into an unthreaded hole, saving much extra effort. They may form the thread by plastic deformation if this is permissible, or else may cut the thread like a tap, in which case they have slots or other openings for the cuttings. Type U metallic drive screws have a large helix angle and are driven in by pressure to make a permanent connection. Wood screws, with a gimlet point and sharp threads, are a kind of thread-forming self-tapping screw. The threads generally extend for 2/3 the length of the screw. The heads are flat, round or oval.

The use of threads for fasteners depends on the action of friction in preventing the loosening of the nut. The fastener can be brought up tight, a small force on the tool reflected in a large force in the screw. However, force in the screw cannot cause the nut to loosen. There are two reasons for this. First, if we consider the thread as an inclined plane of slope φ, a load W on the plane has a component W cos φ normal to the plane, and W sin φ along the plane. If μ is the coefficient of friction, then movement down the plane is impending when W sin φ = μW cos φ, or when μ = tan φ. If tan φ is less than μ, then any load W will not produce motion. For a typical thread, such as 1/4-20 (1/4" major diameter, 20 threads per inch), tan φ = L/πD = 0.064, while for a lubricated metal-metal contact, μ is about 0.1, so there is a factor of safety of about 2 against loosening. Equally important is the friction of the head or nut against the surface of the members joined that prevents rotation.

If φ is sufficiently large, such as 45°, an axial motion of the screw may cause the nut to rotate relatively easily, converting the longitudinal motion into rotational. This arrangment is used in the "Yankee" drills that rotate the bit by pushing the handle of the tool. It is also found in toy tops, that are spun by pumping a handle on the axis.

A washer between the bolt head and the member gives a reliable bearing surface, but may also prevent loosening when designed to do so as a *lock washer*. A spring lockwasher is a spring that exerts a force on the nut even if it is loosened a little, maintaining the pressure and so the frictional resistance. Otherwise, loosening of the nut quickly eliminates this resistance. A toothed lockwasher with teeth around either the inside or the outside, a form often used with machine screws, engages both the nut and the member, preventing any rotation. The nut may be provided with a soft insert of fibre or nylon that grips the bolt and prevents rotation. A *cotter pin* may pass through a slot in the head of a *castle* or *slotted* nut, absolutely preventing rotation. Lockwashers are particularly necessary in the presence of vibration.

Consider a screw of lead L working in a stationary nut. One turn will move the screw through a distance L. Now suppose a second screw of lead L' works in a tapped hole in the first screw, and is prevented from rotating. If both screws are of the same handedness, then one turn moves the outer screw a distance L, while the inner screw moves a distance L' in the other direction, making a net axial movement of L - L'. This is called a *differential screw*, and can be used to produce very small motions per turn without requiring a screw of impracticably small lead. If the screws are of opposite handedness, then the displacements add and we get a rapid motion of L + L' per turn. This is a *compound screw*, of less usefulness than the differential. A *turnbuckle* has threads of opposite senses at its ends, so that a rotation either draws both ends in or out at the same time.

Helical gears are used to connect shafts at an angle to each other. A very important special case is shafts at right angles. Then, a helical *worm* may drive a *worm gear* that can be a normal spur gear, but usually special tooth shapes are used that fit the worm more closely, changing the point contact to a line contact. Worms may be multiple, like screws, to give a more rapid motion. One turn of a single worm corresponds to one tooth on the gear, so the speed ratio is nN, where N is number of teeth on the gear, and n is the multiplicity of the worm. Because of the same frictional factors that prevent the turning of a nut, the worm must drive the gear, not the reverse.

Small-diameter pipe is very conveniently assembled with screwed connections. The U.S. standard pipe thread has the same general thread profile as Unified threads, and can be tapered at 3/4" per foot (1:16) or straight. The tapered threads are designated NPT, while the straight ones are designate NPS. Up to 12", pipe is designated by its nominal internal diameter, while its wall thickness is given by its "schedule number," to which reference must be made to determine the actual outside diameter. Standard threads are 1/16 and 1/8-27, 1/4 and 3/8-18, 1/2 and 3/4-14, then 11-1/2 threads per inch up to 2", and 8 t.p.i. from 2-1/2" upwards. 3/4" pipe, a common small size used in homes, has an O.D. of 1.050".

If a tapered external thread is screwed tightly into a straight thread, the joint is almost, but not quite, leakproof. If American Standard threads are used, there is a fine helical passage between the crests and roots of the threads. For a leakproof joint, a sealing compound must be used in making up the joint. If sealing compound is objectionable, *dryseal* threads may be used instead. For a purely mechanical connection, tapered or straight matching threads may be used, which gives the maximum engagement and strength.

Some topics that can be treated in mechanisms come very close to machines, since they are used to transmit useful forces as well as motion. One example is lifting tackle using pulleys and ropes, which includes interesting differential mechanisms. If velocity is transferred in the ratio v/v', then force is transferred in the ratio f'/f = v/v', if energy is conserved. There is always a close correspondence between velocity and force ratios in a mechanism.

Another such topic is belt drives, which should be called in general *band* drives, where a band is any flexible connector, such as fibre or wire rope, flat belts, V-belts, and chains of many varieties, that can transmit only a tension. Flat belts running on crowned wheels on shafts were once very commonly used to distribute power in a factory, but now electrical subdivision of power is always used, except in very special circumstances, because it is much cheaper and easier to maintain than a magnificent system of shafting. Band drives are still very common as a part of a machine, however, as in automobiles, where they drive the engine accessories as well as, in many cases, the camshafts. Rubber O-rings as band drives are found in many small machines, as a cheap and serviceable means of transferring power, much cheaper than spur gears. Stepped-pulley belt drives are competitive with gears if a range of speeds is required.

There are many useful things to know about band drives, such as the wedging action in V-belt pulleys that gives very positive action, and can also be used with rope drives. Wire rope, however, must not be used in wedging grooves, since this damages it. Wire rope must also not be excessively curved, since any strain beyond the elastic limit will be repetitive, and will soon cause the rope to break. A band connecting shafts at an angle must approach a pulley in a plane perpendicular to the axis of the pulley, but may leave it at an angle. This will prevent the band from leaving the pulley. With small bands, a grooved pulley can overcome some side approach where this is necessary. If you put the band on backwards, it will immediately run off the pulleys.

The effective diameter of a pulley in a band drive is the actual diameter plus the thickness of the band. The correction in adding the thickness of the band is usually negligible, however. Then, speed ratios are the same as diameter ratios. With an *open belt*, the pulleys turn in the same direction; with a *crossed belt* the pulleys turn in opposite directions. My Wimshurst machine uses an open and a crossed O-ring drive on the same shaft to turn the two induction wheels at exactly the same speed.

The linear speed of a band drive is an important design parameter. Flat belts can be used at up to 4500 ft per minute over distances of up to 30 ft. Rope should not be used over 600 ft per minute, but can connect shafts 100 ft apart. Chains are applicable to shorter distances, up to perhaps 15 ft. Roller chain, as used on bicycles, can be used at 2500 ft per minute. The manufacturer's recommendations should be followed in any case. Leather belts should be dressed, and roller belts lubricated.

The power transmitted depends on the speed and on the difference in tensions on the two sides of the pulley. In general, HP = v(T - T')/33,000 where T and T' are the tensions in lb, v the velocity in ft per minute, and HP is in horsepower. Of course, equivalent metric expressions can easily be created.

The difference in tensions when the belt is about to slip is given by T = T'e^{μθ}, where μ is the static coefficient of friction, and θ is the angle of contact in radians. This formula is easily derived by considering a small length of belt in contact with the pulley. It is remarkable in that the radius of the pulley does not enter. The formula is usually derived in the Statics course.

A subject that was treated in the undergraduate mechanisms or kinematics course was what in physics is termed *kinematics*, the analysis of motion without consideration of forces. This is a study of the mathematical relations between displacement s, velocity v and acceleration a in the motion of a particle along a line in one to three dimensions. It is an elementary and concrete subject, one that can be successfully presented in pre-college work, which brings together a number of essential techniques that should be well-known to any engineer. It involves algebraic manipulation, graphical methods, differential and integral calculus, and numerical methods. It can suggest computer programs and provide interesting examples. Moreover, it is of great practical utility. Kinematics is treated at the beginning of every general physics course, as is vector notation. Its review in Kinematics was useful, considering the tendency of engineering students to forget anything learned in Physics. Actually, this is not surprising, since it is really a matter of developing maturity. Learning is essentially an iterative process.

Rotational motion about a fixed axis is analogous to one-dimensional motion along a line, where angle θ corresponds to s, angular velocity ω corresponds to v, and angular acceleration α corresponds to a. The velocity and acceleration of a point on a rotating body is also considered. The more difficult analysis of the kinematics of a rigid body is usually left for advanced courses, but is properly a part of our study here.

It is sufficient to concentrate on rectilinear motion, since motion in a plane is simply the superposition of motions at right angles to each other. Galileo first pointed this out clearly, and it is a fundamental concept, by no means inherently obvious. Acceleration, however, connects motion in different directions. These things are all examples of Galilean Invariance. Motion in a plane can be described by coordinates, such as rectangular or polar. Vectors are only a notational convenience, as all calculations are eventually done with coordinates, but vectors have a conceptual meaning that is also important.

Motion with zero acceleration, or with constant velocity, is simple: the particle moves in a straight line so that s = s_{o} + v_{o}t, where s_{o} is the initial displacement when time t = 0, and v_{o} is the constant velocity. Clearly, v = v_{o} = ds/dt, and s = s_{o} + ∫(t,t_{o})v_{o}dt. These are trivial relations when the acceleration is zero, but give valid results when it is not. They serve to illustrate differentiation and integration.

Motion with a constant specified acceleration a is a very important case, since many motions can be described in terms of intervals where the acceleration is assumed constant. We have a = dv/dt as well as v = v_{o} + ∫(t,t_{o})a dt, which are relations true even when a is not constant. With a = constant, the velocity increases linearly, v = v_{o} + at. The average velocity between two times differing by t is v_{av} = (v_{1} + v_{2})/2, where v_{1} and v_{2} are the velocities at the beginning and end of the interval. By average velocity, we mean that the distance covered is s = v_{av}t. Therefore, s(t) = s_{o} + v_{av}t = s_{o} + v_{o}t + at^{2}, a very familiar expression. The initial displacement s_{o} and the initial velocity v_{o} often make formulas rather complex, so that coordinates are chosen to make them zero, a skill that it is important to learn.

All of these relations between the variables can be made clear by considering the graphical relations shown in the figure. The first three graphs describe motion as a function of the parameter t. Moving to the right corresponds to successive differentiations, and moving to the left successive integrations. These are easily performed by elementary geometrical relations in the case of constant acceleration. For example, the average velocity expression we used in the preceding paragraph is easily derived by considering the area under the v(t) curve, which is a straight line. A trapezoid is replaced by a rectangle of equal area, the height of which is the average of the heights at the two ends, and also the average velocity. Graphical integration and differentiation can be used to construct any of the three graphs if any one of them is known.

This is also a good place to illustrate numerical integration and differentiation, as well as the role of roundoff errors and stability. Numerical integration is efficient and stable, but numerical differentiation is troublesome when the data are not precise. Graphical differentiation, however, is easy because the eye can make adjustments and give a good average slope. When numerically differentiating, it is useful to use a finite minimum time interval and find the average acceleration in that interval. The trouble comes from the limit dt → 0.

The fourth graph in the figure is the result of eliminating the parameter t in the expressions for s(t) and v(t). For constant acceleration, it gives a parabolic variation of v with s. By the elimination of t between v = v_{o} + at and s = s_{o} + v_{o}t + at^{2}, the important relation v^{2} = v_{o}^{2} + 2a(s - s_{o}) is found. If we multiply by m/2, where m is the mass of the particle, then we find (mv^{2}/2) = (mv_{o}^{2}/2) + (ma)(s - s_{o}). Then, if ma = F, the force on the particle, the last term is the *work* done between s and s_{o}, and it is equal to the increase in the value of T = mv^{2}/2, the kinetic energy. This, of course is the principle of conservation of energy, which goes beyond kinematics into dynamics. It shows how kinematics includes little nuggets of imformation that can be used to illustrate important relations.

It should be pointed out that a linear variation of v with s does not correspond to constant acceleration. In fact, if v = ks, then ds/dt = ks, or ds/s = kdt, which integrates to s = s_{o}e^{kt}. Hence s, v and a are all exponential functions of the time, increasing or decreasing depending on the sign of k. For a short time interval Δt, Δv/Δs = a/v, from which the acceleration can be obtained. The analysis of a record of v as a function of s (a "speed tape") is a good practical illustration. Such a record is more common than a record of velocity as a function of time. In a v-s graph, the area under the v(s) curve does not have a useful meaning. The time required to describe a path distance Δs is Δt = Δs/v. This gives good results if the velocity changes little during one step; the step size can be adjusted accordingly. In this way, a v(t) or s(t) curve can be found from a v(s) record.

Motion under gravity is, of course, motion with constant acceleration, making constant acceleration a very practical and useful condition, not just something to make the algebra simple. A freely falling body is a good example. It is convenient to take s_{o} = v_{o} = 0 when t = 0. Then, after falling for a time t and attaining a velocity v = gt, the average velocity is v/2, so that s = (v/2)t = at^{2}/2. Also, from the relation between v and s with constant acceleration, v^{2} = 2gs, or v = √(2gs). If a particle is projected upwards with an intial velocity of v_{o}, then the maximum height reached is given by h = v_{o}^{2}/2g.

In laboratory demonstrations, the most important thing is to establish the constancy of the acceleration g, if not an accurate value for it. There are experments for this purpose, and experiments to find an accurate value for g, and they are not the same. A good value of g can be obtained with a pendulum, but the constancy of acceleration is not shown. A falling body shows constancy of acceleration, but it is not easy to find a good value for g (except with rather advanced electronics). Galileo used an inclined plane to slow down the acceleration, but this brought in rotational motion of the rolling ball. We can now do this more directly on an air track. An Atwood's Machine (shown in the figure) is another excellent demonstration, though not an accurate way to find g. The unbalanced force is (M - m)g and the inertia (M + m), so a = g(M - m)/(M + m).

Galileo's experiment is easy to analyze, since the rotation of a uniform sphere contributes a kinetic energy of (1/5)Mr^{2}, since the kinetic energy is Iω^{2}/2, I = (2/5)Mr^{2}, and v = ωr. Therefore, if h is the distance fallen, then v^{2} = (10/7)gh. The sphere can also be rolled in a concave watch glass, where it will make harmonic oscillations that can be timed, and either g or the radius of the watch glass can be determined.

Distances can be measured by means such as a photograph with stroboscopic illumination, or equally spaced sparks marking a paper strip, or by a movable timing gate on an air track. However this data is obtained, it can be tabulated and differenced. The first differences increase linearly in the ratio 1, 3, 5, 7, ..., as the differences between squares should. The second differences are then constant with the value 2, or really 2a. This can be the basis for a good discussion of numerical and tabular methods.

The two essential illustrations of motion in a plane are uniform motion in a circle, and projectile motion. Uniform motion in a circle illustrates acceleration perpendicular to the velocity, which changes its direction but not its magnitude. The magnitude of this acceleration is the ratio of the change of velocity in one revolution divided by the time to perform one revolution, 2πv divided by 2πr/v, giving a = v^{2}/r. A graphical method of finding this acceleration has been presented in an earlier section. The projection of the motion of the revolving particle on a diameter of the circle is simple harmonic motion, where x = r cos ωt and y = r sin ωt. The acceleration also projects on the diameter, so the acceleration of a point moving on the x-axis with simple harmonic motion is a = - rω^{2}cos ωt = - ω^{2}x, or proportional to the distance from the center of the circle. A spring gives a force F = -kx, so a mass m on a spring k will oscillate with an angular frequency of ω = √(k/m). Again we pass from kinematics to dynamics, with another important example. From this it is a short step to the motion of a simple pendulum.

There is an ingenious way to find centripetal acceleration graphically. If a particle moves uniformly in a circle with speed v around a centre Q, the velocity vector also rotates in a circle with the same angular speed as the point, ω = v/r. The total change in v in one revolution is 2πv, while the time required is 2πr/v. The acceleration is normal to the velocity, directed towards Q, and is of magnitude Δv/Δt = 2πv/(2πr/v) = v^{2}/r, the familiar expression for centripetal acceleration.

On the drawing, let distances be represented according to a scale K, so that an actual distance r corresponds to a distance Kr on the drawing. In the same way, let a velocity v be represented by a distance K'v on the drawing, and an acceleration a by a distance K"a on the drawing.

In the diagram at the right, let AB represent the radius vector or crank, and BC the velocity of point B. Draw a perpendicular from point B to line AC, meeting AC at point D. It is then not hard to show that ΔABC, ΔBCD and ΔABD are all similar; that is, they have the same angles. Then, we have from ΔABC and ΔBCD that CD/BC = BC/AB, or CD = BC^{2}/AB. Now, AB = rK and BC = vK', so CD = (v^{2}/r)K'^{2}/K = a(K'^{2}/K). If we choose K" = K'^{2}/K, then the length of line CD will give the centripetal acceleration to scale. In general, we can choose K and K' in any manner, and then determine K" from them, which can be used any place on the drawing to find the centripetal acceleration.

The comprehension of how certain linkages work is made much easier if the operation of a model is observed. Such models are very easily made using stiff cardboard. I used some old shirt cardboards that I had around, that have one side with a white surface. The stouter the cardboard the better, up to the point where it becomes hard to work. Links are made from 3/4" (20 mm) strips cut with a cutting board. Joints are made from split brass-colored fasteners of the 1" size (OIC #99814), available at 100 for a dollar. A leather punch is used to make 1/8" (3 mm) holes for the joints between links. Fixed joints are made with a small slit cut with a pointed X-acto knife or a single-edge razor blade in the sheet of cardboard that forms the frame. A scale will be needed to make measurements. Before punching the holes in the links, mark the centre line and the hole locations. Use smaller holes to mark points that are not joints, such as places where a pencil can be inserted to draw a line. Although these models are not suitable for heavy use, they show the motions very well indeed.

P. Schwamb, A. L. Merrill and W. H. James, *Elements of Mechanism*, 6th ed. (New York: John Wiley & Sons, 1947). A classic text on graphics-based mechanisms.

S. P. Parker, ed., *McGraw-Hill Encyclopedia of Engineering*, 2nd. ed. (New York: McGraw-Hill, 1992). Art. "Mechanism," pp. 697-700.

V. M. Faires, *Design of Machine Elements*, 4th ed. (New York: Macmillan, 1965). Chapters 13-16 of this excellent text treat gears, and even explain the Law of Gearing, but only involute gearing is mentioned.

C. Stoll, *The Curious History of the First Pocket Calculator*, *Scientific American*, **290**(1), January 2004. p. 92.

R. L. Hills, *Power from Steam* (Cambridge: Cambridge University Press, 1989). An excellent source for the history of the steam engine. An illustration of a Watt rack-and-sector straight line motion is on p. 67, and a good drawing of the pantograph extensions on p. 68. Draw a line from the main bearing to the crosshead, and it will pass through the connection to the air pump rod. The central joint on the centre link is not the point that is guided in a straight line, but an unmarked point below it, where the line you have drawn crosses it.

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Composed by J. B. Calvert

Created 24 December 2003

Last revised 2 December 2005