Kinematics of machines - Durley

KINEMATICS OF MACHINES

AN ELEMENTARY TEXT-BOOK

BY R. J. DURLEY

NEW YORK, JOHN WILEY & SONS, 1911

Kinematics of machines

CHAPTER I - INTRODUCTORY CONSIDERATIONS

1. Study of Machines - In general the study of a Machine involves problems of three distinct kinds. We may first of all consider from a geometrical point of view the motion of any part of the machine with reference to any other part, without taking account of any of the forces acting on such parts. Or, the action of the forces impressed on the parts of the machine, and of the forces due to its own inertia or to the weight of its parts, may be dealt with, and the resulting transformations of energy may be determined. A third branch of the theory of machines treats of the action of these loads and forces in producing stresses and strains in the materials employed in the construction of the machine, and discusses the sizes, forms, and pro- portions of the various parts which are required either to insure proper strength while avoiding waste of material, or to make the machine capable of doing the work for which it is being designed.

The science dealing with the first-named class of problem is termed the Kinematics of Machines, which we may define as being that science which treats of the relative motion of the parts of machines, without regard to the forces producing such motions, or to the stresses and strains produced by such forces. With this limitation, in the case of almost all bodies forming portions of machines, it is possible to neglect any deformation they may undergo in working, and in studying the Kinematics of Machines we may at once apply to machine problems the results obtained by the study of the motion of rigid bodies. Important exceptions will present themselves to the reader's mind; for example, ropes, belts, and springs cannot be considered kinematically as being rigid, and many mechanical contrivances involve the use of liquid or gaseous material. Such cases as these will be considered later.

By the term Machine we may understand a combination or arrangement of certain portions of resistant material, the relative motions of which are controlled in such a way that some form of available energy is transmitted from place to place, or is transformed into another desired kind. This definition includes under the head of Machines all contrivances which have for their object the transformation or transmission of energy, or the performance of some particular kind of work, and further implies that a single portion of material is not considered as a machine. The so-called simple machines in every case involve the idea of more than one piece of material.

A combination or arrangement of portions of material by means of which forces are transmitted or loads are carried without sensible relative motions of the component parts is called o. Structure.

The term Mechanism is often used as an equivalent for the word Machine. It is, however, preferable to restrict its use somewhat, and to employ the word to denote simply a combination of pieces of material having definite relative motions, one of the pieces being regarded as fixed in space. Such a mechanism often represents kinematically some actual machine which has the same number of parts as the mechanism with the same relative motions. The essential difference is that in the case of a machine such parts have to transmit or transform energy, and are proportioned and formed for this end, while in a mechanism the relative motion of the parts only is considered. We may look upon a mechanism, then, as being the ideal or kinematic form of a machine, and our work will be much simplified in most cases if we consider for kinematic purposes the mechanism instead of the machine. Such a substitution is also of the greatest service in the comparison and classification of machines; we shall find in this way that machines, at first sight quite distinct, are really related, inasmuch as their representative mechanisms consist of the same number of parts having similar relative motions, and only differing because a different piece is considered to be fixed in each case.

2. Constrained Motion - On further consideration of the nature of a Machine as defined above, it will be noted that each part of the machine must have certain definite motions relatively to any other part, such definite motions being repeated again and again during the working of the machine. Thus the motion of a machine-part must be completely constrained, that is, the part must be free to move only in the manner desired to produce the required transformation of energy, and for it other unnecessary motions must be rendered impossible. Constrained motion of a body takes place when every point in the body is made to describe some definite and prescribed path. This constraint is effected in general by so forming and connecting the parts that all forces tending to disturb their constrained motion are balanced by stresses set up in the parts themselves. It is assumed, of course, that the machine remains uninjured by such stresses.

6. Non-plane Motion - In the majority of cases it will be found that the relative motions of the parts of machines are plane motions, either of rotation or translation, or both combined. Such motions can be studied geometrically by the method indicated in the preceding section. It is possible (as will be seen later) to have a lower pair, in which the motion is non-plane. A somewhat limited number of cases of higher pairing also occur in which the motion is non-plane.

In every instance, however, in a closed pair, we have seen that there must be continuous contact of the surfaces, and it follows that the most general possible relative motion of two parts of a mechanism is represented by the motion of one rigid body continuously touching another at a point or series of points.

Any such motion must be of the nature of sliding, rolling, or spinning, separately or combined. Simple rolling takes place if the instantaneous axis lies in the common tangent plane at the point of instantaneous contact.

Simple spinning exists when the instantaneous axis is the common normal at the point of contact.

Suppose that the relative motion is such that the instantaneous axis passes through the point of contact, and is neither in nor perpendicular to the tangent plane. The motion is then combined rolling and spinning. If the instantaneous axis does not pass through the point of contact, the rolling and spinning will further be combined with a sliding motion.

We have a familiar example of combined rolling and sliding in the mutual action of a pair of teeth in an ordinary spur-wheel; the motion of the balls in a bicycle bearing, again, is a case of combined rolling and spinning.

The links of a certain class of mechanism are found to have such motions that their instantaneous axes all pass through a fixed point, while each portion of every link remains at its own constant distance from that point. Such motion is called spheric motion, because any given point on a link must be always on the surface of a sphere described about the fixed point as centre. It is evident that the most general case of spheric motion is that of a rigid body of which one point is fixed, and any kind of spheric motion can be made up by combining spins about axes passing through the fixed point. Plane motion may be looked upon as a particular case of spheric motion, in which the radius of the spheres is infinitely large.

7. Freedom and Constraint - We have seen that the essential feature of a kinematic pair is the mutual constraint due to the forms of the two elements of which the pair is composed. Before considering the ways in which constraint or closure is actually applied it will be well to examine briefly the conditions on which the freedom of movement of a rigid body depends.

The most general motion of a free rigid body may be looked upon as being a combination of three independent rotations about three rectangular axes, with three independent motions of translation along those axes. Such a body may then be said to have six degrees of freedom one of which is taken away (or one degree of constraint is imposed) when any one of these six modes of movement is rendered impossible. Suppose that the free rigid body is forced to touch a smooth fixed surface at one point, one degree of freedom is lost, for no translation can take place in a direction normal to the tangent plane to the surface at the points of contact. The three motions of rotation, however, still remain possible, and so does motion of translation in any direction parallel to the tangent plane at the point of contact. A second point of restraint may be arranged so as to prevent one motion of rotation, or a second motion of translation, according to its position with regard to the first point of restraint and with regard to the form of the body. A third point of restraint causes the body to lose a third degree of freedom, and, finally, it will be found that all six degrees of freedom are lost, and the position of the body is fixed if six of its points are made to rest on six portions of the surface of the smooth fixed body, and if these portions are properly formed and placed.

It may be shown that in general six conditions are required to completely determine the position of a rigid body, or, expressing the same thing in another way, six coordinates specify the position of one rigid body relatively to another, considered to be fixed.

Consider, for example, a screw turning in a fixed nut, like the screw of a micrometer gauge. The position of such a screw is determined exactly if an arm attached to its head is forced to remain in contact with a fixed stop on the body of the gauge, and we say, therefore, that such a screw has only one degree of freedom, inasmuch as its position is fixed by one point of constraint. The motion of a screw in its nut, a motion of translation accompanied by a definite and proportional motion of rotation whose axis is the direction of translation, is the most general kind of motion that can be possessed by a body having only one degree of freedom.

The reader will notice that in two special cases, namely, when the pitch of the screw is infinite, and when the pitch is zero, the twisting motion of the nut becomes a mere translation or a mere rotation, both being specially important as plane motion involving one degree of freedom.

In a similar way such a body as the connecting-rod of a direct-acting steam-engine is said to have constrained motion, having only one degree of freedom. The only possible motion at any instant for a given point on the rod is that of rotation about a certain virtual axis parallel to the axis of the crank-shaft.

CONTENTS.

- INTRODUCTORY CONSIDERATIONS
- POSITION, VELOCITY, AND ACCELERATION
- PLANE MECHANISMS CONTAINING ONLY TURNING PAIRS
- SLIDER-CRANK CHAINS
- DETERMINATION OF VELOCITY AND ACCELERATION IN PLANE MECHANISMS
- ALTERATION OF MECHANISMS. CLOSURE.
- CONSTRAINT AND VELOCITY RATIO IN HIGHER PAIRING INVOLVING PLANE MOTION.
- WHEEL-TRAINS AND MECHANISMS CONTAINING THEM. CAMS.
- RATCHET MECHANISMS AND ESCAPEMENTS