q be the synchronal rotations of two wheels whose angular velo

cities are a and a, respectively; then 2 =

A

q a

; that is, synchronal

rotations are in the ratio of the angular velocities.

Example.-Let a wheel whose radius is 6 ft. perform 50 revolutions per min., required 1st, the velocity of its circumference, and 2nd, its angular velocity.

Here, by eq. (1), n=50, and r=6, then

1

v = × 3.1416 × 50 × 6 = 31·416 ft. per sec.

30

1

And, by eq. (4), a = × 3.1416 × 50 = 5·236.

30

8. If v and v be the velocities of two parts of a piece of mechanism, then is the velocity ratio of these parts. Let s and

ข

8 be the corresponding spaces described in the same time, then when the motion is uniform

that is, when the velocities are uniform, the velocity ratio is constant.

9. If the velocity ratio of the two parts remains constant, then, however variable the velocities themselves may be, we still shall have = ; where s and 8 are the entire spaces de

V
ย

8

scribed in the same interval of time.

at

10. When a body moves with a variable motion, its velocity any instant is determined by the rate at which it is moving at that particular instant, that is, by the space which it would move over in one second, supposing the motion which it then has to remain constant for that time. Variable motions may be graphically represented, by taking the abscissa of a curve equal to the units of time, and the ordinates equal to the units of the corresponding velocities. Thus, let A B be equal to the units of velocity at the commencement of the motion; A c the units in

D

R

Fig. 3.

A

X

interval of time; C D the units in the corresponding velocity; and so on; then the area of the curved space A B D F E will be equal to the space described in the interval of time represented by A E.

If the motion be uniform, the curve BDF will become a straight line parallel to ox, and the space described in any given time will be represented by the area of a rectangle, whose length is equal to the units of time, and breadth equal to the units of velocity.

If the motion be uniformly accelerated or retarded, the curve B D F will become a straight line inclined to the axis o x, and the space described, in this case, will be represented by the area of a trapezoid, whose base is equal to the units of time, and parallel sides respectively to the velocity at the commencement and end of that time.

11. THE PARTS OF A MACHINE.-A machine consists of three important parts.

(1.) The parts which receive the work of the moving power— these may be called RECEIVERS of work.

(2.) The parts which perform the work to be done by the machine—these may be called WORKING PARTS, or more simply,

OPERATORS.

(3.) The mechanism which transmits the work from the receivers to the working parts or operators-these pieces of mechanism may be called COMMUNICATORS OF WORK, or the

TRANSMISSIVE MACHINERY.

The form of the mechanism must always be determined from the relation subsisting between the motions of the receivers and operators.

If there were no loss of work in transmission (from friction, &c.) the work applied to the receiver would always be equal to the work done by the operator. Thus, let P be the lbs. pressure applied to the receiver, and s the space in feet which it moves over in a certain time; P1 the lbs. pressure produced at the working part, and s, the space in feet which it moves over in the same time; then, neglecting the loss of work by friction, we have

=

Work applied to the receiver work done upon the operator, or P X SP1 X S1 (1).

1

However, it must be borne in mind, that the actual or useful work done by a machine is always a certain fractional part of the work applied; this fraction, determined for any particular machine, is called the modulus of that machine. If m be put for this modulus, then we have from eq. (1)

In treating of the motion of these parts of a machine it is generally most convenient to find an expression for their proportional velocities. Thus, let v be the velocity of the receiver, and v1 that of the operator; then is their velocity ratio. See

Art. 8.

V

It must be observed, that this velocity ratio is not at all effected by the actual velocities of the parts, provided the velocity ratio of the mechanism be constant for all positions. In the more ordinary pieces of mechanism (such as common toothed wheels, wheels moved by straps, levers, &c.) the velocity ratio is constant; that is to say, it remains the same for all positions of the mechanism.

In eq. (1) s may be taken as the velocity of the power P, estimated in the direction in which it acts, and s, that of the resistance P1; then this equality becomes

Now is called the advantage gained by the machine, or the

Р

number of times that the resistance moved is greater than the power applied. Hence the advantage gained by a machine, irrespective of friction, &c., is equal to the velocity of the power divided by the velocity of the resistance, or the velocity ratio of the power and resistance.

This is called the principle of virtual velocities. Workmen express this dynamic law by saying, 'What is gained in power is lost in speed.'

12. The DIRECTIONAL RELATION of the motion of the receiver and the operator admits of every possible variation. It may be constant or it may be variable. By the intervention of me

chanism rectilinear motion may be converted into curvilinear motion, and conversely; reciprocating rectilinear or circular motion may be converted into continuous circular motion, and conversely; and so on to the various possible combinations of which the cases admit. These directional changes are so important, in a practical point of view, that some eminent writers on mechanism have made them the basis of the classification of mechanism. But, however eligible in a practical point of light such a classification may be, there is complexity in its application, which renders it less suitable for scientific purposes than that method of classification which is based upon the nature or mode of action of certain elementary pieces of mechanism which enter, more or less, into every mechanical combination.

Elementary forms of Mechanism.

13. In analysing the parts of a machine we find motion transmitted by jointed rods or links, by straps and cords, by wheels rolling on other wheels, and by pieces of various forms sliding or slipping on other pieces. Hence we have the following elementary forms of mechanism :

(1.) Transmission of motion by jointed rods,-LINK-WORK. (2.) By straps, cords, &c.,-WRAPPING CONNECTORS.

(3.) By wheels or curved surfaces, revolving on centres, rolling on each other,-WHEEL-WORK.

(4.) By pieces of various forms, sliding or slipping on each other, SLIDING-PIECES.

14. The velocity ratio, as well as the directional relation, in an elementary piece of mechanism may be either constant or varying. The number of combinations of which these elementary pieces admit, is almost unlimited. The eccentric wheel is a combination of sliding pieces and link-work. The common crane is a combination of wheel-work, link-work, and wrapping connectors; and so on to other cases.

A train of mechanism must be supported by some frame work; the train of pieces being such, that when the receiver is moved the other pieces are constrained to move in the manner determined by the mode of their connection. Revolving pieces, such as wheels and pulleys, are so connected with the frame that

every portion of them is constrained to move in a circle round the axis; and sliding pieces are constrained to move in straight lines by guides.

Mechanism is to a great extent a geometrical inquiry. The motion of one piece in a train may differ, both in kind and direction, from the motion of the next piece in the series: these changes are effected by the geometrical construction of the pieces, as well as by their mode of connection. The investigation of the law of these changes constitutes one of the chief objects of the principles of mechanism.

II. ON LINK-WORK.

15. If a bent rod or lever a C B turn upon the centre c, the velocities of the extremities A and B will be to each other in the ratio of their distances from the centre of motion c; that is,

It is not necessary that the arms A C and B C should be in the same plane. Thus let C D be an axis round which the arms A E and B F revolve, then,