This book is the outcome of several years' experience in teaching kinematics and kinetics of machinery at the University of Illinois. For many years this subject was taught from notes prepared by Professor G. A. Goodenough, to which was added an article on the gyroscope by Professor F. B. Seely of the Department of Theoretical and Applied Mechanics. These notes were several times revised by the authors as experience showed where improvements could be made.

In the fall of 1916 the authors undertook, with the consent of Messrs. Goodenough and Seely, to rewrite these notes in textbook form. The present volume is the outcome of that undertaking. The work was interrupted by the war, which took one of the writers into the military service, and imposed on the other such a heavy burden of teaching work that further progress on the book was impossible. In the fall of 1919 the work was resumed and pushed to completion.

The introductory chapter on constraint, the chapter on plane motion, and the chapter on velocities follow closely Professor Goodenough's notes. The chapter on the gyroscope is inserted almost without change as written by Professor Seely. The chapters on accelerations, on inertia forces, on balancing and on governors have been so completely rewritten that little trace remains of the original. The chapters on toothed wheels, on cams, on wrapping connectors, and on critical speeds have been added by the authors.

It is hoped that this volume will fill a need in the curricula of our engineering schools, in that it gives systematic methods of determining velocities, accelerations, and inertia forces which can be applied to practically all mechanisms. These methods are in the main graphical; the complicated forms of the equations making analytical methods too cumbersome for practical use except in some of the simpler types of machines. If the work is done to a large scale the results should be accurate enough for all practical purposes.

The book is so arranged that it can be readily adapted to short courses as well as to more complete and detailed ones. Thus the chapters on gears, cams, and belts may be omitted where these subjects are taught in the courses in mechanism or design. The chapter on balancing can be profitably studied without the detailed analysis given in the chapters on accelerations and inertia forces. The chapter on critical speeds and parts of the chapters on governors and gyroscopes involve the use of mathematics which is perhaps beyond the range of the average undergraduate. These parts may, however, be of great value to the advanced student who intends to specialize in scientific design. For the benefit of undergraduate students a note on the solution of linear differential equations is appended.

In conclusion the authors wish to extend their thanks to Professor Goodenough for valuable suggestions and criticisms in the preparation of the work.

1. Scope of the Subject. Mechanics is that branch of Physics which treats of the motions of material bodies and the forces acting on such bodies. That division of mechanics which deals with motions is called kinematics, and that division which deals with forces is called dynamics. Dynamics is further divided into statics, in which the bodies dealt with are considered to be in equilibrium, and kinetics, hi which the bodies are acted upon by unbalanced forces.

Mechanics of machinery consists of the study of the application of the laws of mechanics to the parts of machines. The subject may be divided again into kinematics, statics, and kinetics. Kinematics of machinery consists of the study of the motions of the parts of machines without regard to the forces accompanying or producing such motions. Statics of machinery consists of the study of the forces in machine parts with the assumption that all such parts are in equilibrium, that is, disregarding any forces which may act to produce accelerations in these bodies.

In any machine it is impossible that any part shall move indefinitely at a uniform speed and in a straight line. It follows that there must always be forces producing accelerations in the moving parts, or in other words that no moving part can be in equilibrium. The province of kinetics of machinery is to take into account the accelerating forces. In many instances the accelerating forces are quite small, and the problems arising in such cases may be treated by static methods. In other cases the accelerating forces are extremely important. This is true of all high-speed machinery. In the following pages the subjects of kinematics and kinetics of machinery will be treated at length. Static problems will be considered only incidentally.

The study of mechanics of machinery may be approached from two different points of view: (1) the motions and forces in existing machines may be analyzed; and (2) machines may be devised to produce desired motions and forces. It is believed that a thorough study and analysis of existing machines will be of great value to those who later expect to become designers, and it is the purpose of this book to guide students in such study.

In general, a machine will be regarded as a system of rigid bodies, so connected that for a given movement of any one part there will be perfectly definite, determinate movement of every other part. The assumption of the rigidity of the parts is equivalent to disregarding any motions due to distortion or vibration of the members. Special cases where such distortions are not negligible, or where flexible links such as belts and ropes are employed, will be given special attention.

2. Constrained Motion. Pairs. The characteristic of the motion of a machine part, as distinguished from that of a free body, is that every point of the machine element is constrained to move in a fixed predetermined path. In order that this may be the case, it is necessary that each machine part must be in contact with one or more other parts. The connections between the parts are called pairs. Thus in the ordinary steam engine mechanism, Fig. 1, there are four pairs: (1) between the crank and bearing, (2) between the connecting rod and crank, (3) between the cross-head and connecting rod, and (4) between the crosshead and guides.

3. Properties of Pairs. Since the purpose of pairs is to constrain the relative motion between the pairing bodies, the first step in analyzing constrained motions is the study of the properties of pairs. The simplest way to constrain a point P to move in a given path would be to cut a slot whose center line is the given path, and place in the slot a block on which is marked the point to be guided. In general, if this block is cut to fit the curvature of the slot in one position, it will not fit in some other position where the curvature of the slot is different, Fig. 2. If, however, the radius of curvature of the slot is constant, the block will fit equally well in all positions, Fig. 3. If the radius of curvature of the slot is indefinitely increased the circle becomes a straight line, Fig. 4. Rectilinear motion may therefore be regarded as a limiting case of circular motion. It follows that continuous surface contact between pairing bodies is possible only when the relative motion is circular or rectilinear. For paths which have variable curvature only line contact is possible. For example, if the sliding block, Fig. 2, is replaced by a circular pin, Fig. 5, relative motion becomes possible regardless of the curvature of the slot. Pairs which permit surface contact are called lower pairs. Those which permit only line contact are called higher pairs.

4. Pairing Elements. The geometrical forms placed upon two bodies so that they may be connected by a pair are called pairing elements. Thus the cylindrical surface of the wrist pin of an engine, and the inside surface of the brasses of the connecting rod are pairing elements. The surfaces of the crosshead and guides, the faces of two gear teeth in mesh, or of two cams in contact, furnish other examples.

5. Inversion of Pairs. In the case of lower pairs the solid and hollow elements may be interchanged without changing the character of the relative motion. For example, instead of having a pin in the crosshead of an engine fitting into an eye in the connecting rod, a pin might be attached to the rod fitting into a hole in the crosshead. Instead of having a moving piston in a stationary cylinder a moving cylinder might slide over a fixed piston, the connecting rod being attached to the moving cylinder as shown in Fig. 11. The process of exchanging the hollow and solid elements of a pair is called inversion of the pair.

6. Lower Pairs with Multiple Contact. It frequently occurs in machine construction that a single lower pair may have several contact surfaces. Thus the shaft of an engine runs in two bearings, and both the crosshead and piston are provided with sliding pairs. A single sliding pair may have any number of contact surfaces provided these surfaces are parallel, and a single turning pair may have any number of journals and bearings provided that they all have the same axis.

7. Links. A body provided with two or more pairing elements, so that it may be connected to at least two other bodies, is called a kinematic link, or simply a link. A body with two pairing elements is called a binary link, one with three pairing elements a ternary link, one with four pairing elements a quaternary link.

In machines having more than two members each link must have at least two pairing elements in order that the motions of all links may be constrained. It should be noted that the term link applies to all parts of a machine which are rigidly fastened together so that there can be no relative motion. For example, in the steam engine, Fig. 1, we may identify the following links:

1. Cylinder and cylinder heads, base plate, foundation, main bearing, crosshead guides, etc.

2. Piston and rings, piston rod, crosshead, wedges, bolts, etc.

3. Connecting rod, brasses, straps, wedges, etc.

4. Crank shaft, crank arm, crank pin, counterweight, fly wheel, etc.

The motions in the machine depend entirely on the relative positions of the pairs, and not at all on the size and shape of the links. In particular the stationary link may be very large and heavy, and extra material may be added indefinitely, without in any way affecting the motions, so long as the positions of the pairing elements are unchanged.

8. Chains. If a number of links are provided with suitable pairing elements they may be connected by joining the elements. If this is done so that each element is provided with a mate and none left unpaired, the resulting structure is called a kinematic chain, or simply a chain. Such chains are divided into three classes as follows:

1. Constrained chains, in which relative motions of the links are possible, and where such motions are completely predetermined by the character of the pairs.

2. Locked chains, in which no relative motion is possible.

3. Unconstrained chains, in which the relative motions are indeterminate.

In Figs. 20 to 22 are shown chains consisting of three, four, and five binary links respectively. In Fig. 20 evidently no relative motion is possible and the chain is therefore locked. Thus if link 1 is held fast, say to the paper, the axis of the pair B considered as part of link 2 must move, if at all, in an arc about A as a center. Likewise considering the axis of B as a point on link 3 it must move, if at all, in an arc about C as a center. But, since the point cannot move simultaneously in two different paths, it cannot move at all, and the chain is therefore locked.


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