CHAPTER II. ON APPLIED DYNAMICS
§ 83. Laws of Motion.The action of a machine in transmitting force and motion simultaneously, or performing work, is governed, in common with the phenomena of moving bodies in general, by two " laws of motion."
Division 1. Balanced Forces in Machines of Uniform Velocity.
§ 84. Application of Force to Mechanism.—Forces are applied in units of weight; and the unit most commonly employed in Britain is the pound avoirdupois. The action of a force applied to a body is always in reality distributed over some definite space, either a volume of three dimensions or a surface of two. An example of a force distributed throughout a volume is the weight of the body itself, which acts on every particle, however small. The pressure exerted between two bodies at their surface of contact, or between the two parts of one body on either side of an ideal surface of separation, is an example of a force distributed over a surface. The mode of distribution of a force applied to a solid body requires to be considered when its stiffness and strength are treated of ; but, in questions respecting the action of a force upon a rigid body considered as a whole, the resultant of the distributed force, determined according to the principles of statics, and considered as acting in a single line and applied at a single point, may, for the occasion, be substituted for the force as really distributed. Thus, the weight of each separate piece in a machine is treated as acting wholly at its centre of gravity, and each pressure applied to it as acting at a point called the centre of pressure of the surface to which the pressure is really applied.
§ 85. Forces applied to Mechanism Classed.—If B be the obliquity of a force F applied to a piece of a machine—that is, the angle made by the direction of the force with the direction of motion of its point of application—then by the principles of statics, F may be resolved into two rectangular components, viz.
Along the direction of motion, P=F cos 0 (49f Across the direction of motion, Q =F sin B
C
x
B
If the component along the direction of motion acts with the motion, it is called an effort; if against the motion, a resistance. The component across the direction of motion is a lateral pressure; the unbalanced lateral pressure on any piece, or part of a piece, is deflecting force. A lateral pressure may increase resistance by causing friction; the friction so caused acts against the motion, and is a resistance, but the lateral pressure causing it is not a resistance. Resistances are distinguished into useful and prejudicial, according as they arise from the useful effect produced by the machine or from other causes.
§ 86. Work.—Work consists in moving against resistance. The work is said to be performed, and the resistance overcome. Work is measured by the product of the resistance into the distance through which its point of application is moved. The unit of work commonly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called a footpound.
Work is distinguished into useful work and prejudicial or lost work, according as it is performed in producing the useful effect of the machine, or in overcoming prejudicial resistance.
§ 87. Energy: Potential Energy.—Energy means capacity for performing work. The energy of an effort, or potential energy, is measured by the product of the effort into the distance through which its point of application is capable of being moved. The unit of energy is the same with the unit of work.
When the point of application of an effort has been moved through a given distance, energy is said to have been exerted to an amount expressed by the product of the effort into the distance through which its point of application has been moved.
§ 88. Variable Effort and Resistance.—If an effort has different magnitudes during different portions of the motion of its point of application through a given distance, let each different magnitude of the effort P be multiplied by the length As of the corresponding portion of the path of the point of application; the sum
I. POs (5o) is the whole energy exerted. If the effort varies by insensible gradations, the energy exerted is the integral or limit towards which that sum approaches continually as the divisions of the path are made smaller and more numerous, and is expressed by
f Pds. (51)
Similar processes are applicable to the finding of the work performed in overcoming a varying resistance.
The work done by a machine can be actually measured by means of a dynamometer (q.v.).
§ 89. Principle of the Equality of Energy and Work.—From the first law of motion it follows that in a machine whose pieces move with uniform velocities the efforts and resistances must balance each other. Now from the laws of statics it is known that, in order that a system of forces applied to a system of connected points may he in equilibrium, it is necessary that the sum formed by putting together the products of the forces by the respective distances through which their points of application are capable of moving simultaneously, each along the direction of the force applied to it, shall be zero,—products being considered positive or negative according as the direction of the forces and the possible motions of their points of application are the same or opposite.
In other words, the sum of the negative products is equal to the sum of the positive products. This principle, applied to a machine whose parts move with uniform velocities, is equivalent to saying that in any given interval of time the energy exerted is equal to the work performed.
The symbolical expression of this law is as follows: let efforts be applied to one or any number of points of a machine; let any one of these efforts be represented by P, and the distance traversed by its point of application in a given interval of time by ds; let resistances be overcome at one or any number of points of the same machine; let any one of these resistances be denoted by R, and the distance traversed by its point of application in the gi en interval of time by ds'; then
. Pds=E . Rds'. (52)
The lengths ds, ds' are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I.
§ 9o.* Static Equilibrium of Mechanisms.—The principle stated in the preceding section, namely, that the energy exerted is equal to [he work performed, enables the ratio of the components of the forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found. Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds' were made, it becomes Pds/dt = Rds'/dt, that is, Pv = Rv', which shows that the force ratio is the inverse of the velocity ratio. It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the directionof motion of the points. The relation between the effort and the resistance may be found by means of this principle for all kinds of mechanisms, when the friction produced by the components of the forces across the direction of motion of the two points is neglected. Consider the following example:
A fourbar chain having the configuration shown in fig. 126 supports a load P at the point x. What load is required at the point y to maintain the con
figuration
shown, both loads being supposed to act vertically? Find the instantaneous centre Obd, and resolve each load in the respective directions of motion of the points x and y; thus there are obtained the components P cos B and R cos ¢. Let the mechanism have a small motion; then, for the instant, the link b is turning about its
instantaneous centre Obd, and, if co is its instantaneous angular velocity, the velocity of the point x is wr, and the velocity of the point y is cos. Hence, by the principle just stated, P cos B X cer =R cos di Xcos. But, p and q being respectively the perpendiculars to the lines of action of the forces, this equation reduces to Pp = Rq, which shows that the
ratio of the two forces may be found by taking moments about the instantaneous centre of the link on which they act.
The forces P and R may, however, act on different links. The general problem may then be thus stated: Given a mechanism of which r is the fixed link, and s and t any other two links, given also a force ,t', acting on the link s, to find the force ft acting in a given direction on the link t, which will keep the mechanism in static equilibrium. The graphic solution of this problem may be effected thus:
(1) Find the three virtual centres O,.,, Ore, O,t, which must be three points in a line.
(2) Resolve f, into two components, one of which, namely, f5, passes through 0,,, and may be neglected, and the other fp passes through O,t.
Find the point M, where f5 joins the given direction of ft, and resolve fp into two components, of which one is in the direction MO,.t and may be neglected because it passes through Orr, and the other is in the given direction of ft and is therefore the force required.
This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is given in ch. 8 of his Mechanics of Machinery. Another general solution of
the problem is given in the Proc. Lond. Math. Soc. (18781879), by the same author. An example of the method of solution stated above, and taken from the Mechanics of Machinery, is illustrated by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r fixed. Let it be required to find the vertical force which must act at the pitch radius of the last wheel t to balance exactly a force f, acting vertically downwards on the arm at the point indicated in the figure. The two links concerned are the last wheel t
and the arm s, the wheel r being the fixed link of the mechanism. The virtual centres Or„ Oat are at the respective axes of the wheels r and t, and the centre Ort divides the line through these two points externally in the ratio of the train of wheels. The figure sufficiently indicates the various steps of the solution.
The relation between the effort and the resistance in a machine to include the effect of friction at the joints has been investigated in a paper by Professor Fleeming Jenkin, " On the application of graphic methods to the determination of the efficiency of machinery "
(3)
(Trans. Roy. Soc. Ed., vol. 28). It is shown that a machine may at any instant be represented by a frame of links the stresses in which are identical with the pressures at the joints of the mechanism. This selfstrained frame is called the dynamic frame of the machine. The driving and resisting efforts are represented by elastic links in the dynamic frame, and when the frame with its elastic links is drawn the stresses in the several members of it may be determined by means of reciprocal figures. Incidentally the method gives the pressures at every joint of the mechanism.
§ 91. Efficiency.—The efficiency of a machine is the ratio of the useful work to the total work—that is, to the energy exerted—and is represented by
R„ ds' . R„ ds' Z. Ruds' U . Rds' =E . R.ds'+E . R,dsl= E . Pds =E.
E = (1 +A)U+B, (55)
where A and B are constants, depending on the form, arrangement and weight of the mechanism. The efficiency corresponding to the
last equation is U
E = I (56)
1 +A+B/U
§ 94. Trains of Mechanism.—In applying the preceding principles to a train of mechanism, it may either be treated as a whole, or it may be considered in sections consisting of single pieces, or of any convenient portion of the train—each section being treated as a machine driven by the effort applied to it and energy exerted upon it through its line of connexion with the preceding section, performing useful work by driving the following section, and losing work by overcoming its own prejudicial resistances. It is evident
that the efficiency of the whole train is the product of the efficiencies of its sections.
§ 95. Rotating Pieces: Couples of Forces.—It is often convenient to express the energy exerted upon and the work performed by a turning piece in a machine in terms of the moment of the couples of forces acting on it, and of the angular velocity. The ordinary British unit of moment is a footpound; but it is to be remembered that this is a footpound of a different sort from the unit of energy and work.
If a force be applied to a turning piece in a line pot passing through its axis, the axis will press against its bearings with an equal and parallel force, and the equal and opposite reaction of the bearings will constitute, together with the firstmentioned force, a couple whose arm is the perpendicular distance from the axis to the line of action of the first force.
A couple is said to be right or left handed with reference to the observer, according to the direction in which it tends to turn the body, and is a driving couple or a resisting couple according as its tendency is with or against that of the actual rotation.
Let dt be an interval of time, a the angular velocity of the piece; then adt is the angle through which it turns in the interval dt, and ds =vdt=radt is the distance through which the point of application of the force moves. Let P represent an effort, so that Pr is a driving couple, thencouple, which, if applied to the other point or piece, would exert equal energy or employ equal work. The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.
These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed.
§ 97. Balanced Lateral Pressure of Guides and Bearings.—The
most important part of the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included. There may be or may not be an unbalanced component in this pressure, due to the deviated motion. Its laws will be considered in the sequel.
§ 98. Friction. Unguents.—The most important kind of resistance in machines is the friction or rubbing resistance of surfaces which slide over each other. The direction of the resistance of friction is opposite to that in which the sliding takes place. Its magnitude is the product of the normal pressure or force which presses the. rubbing surfaces together in• a direction perpendicular to themselves into a specific constant already mentioned in § 14, as the coefficient of friction, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered. The total pressure exerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and its obliquity, or inclination to the common perpendicular of the surfaces, is the angle of repose formerly mentioned in § 14, whose tangent is the coefficient of friction. Thus, let N be the normal pressure, R the friction, T the total pressure, f the coefficient of friction, and ¢ the angle of repose; then
f = tan 4, (58) R=fN=N tan ¢=T sin ¢
Experiments on friction have been made by Coulomb, Samuel Vince, John Rennie, James Wood, D. Rankine and others. The most complete and elaborate experiments are those of Morin, published in his Notions fondamentales de mecanique, and republished in Britain in the works of Moseley and Gordon.
The experiments of Beauchamp Tower (" Report of Friction Experiments," Proc. Inst. Mech. Eng., 1883) showed that when oil is supplied to a journal by means of an oil bath the coefficient of friction varies nearly inversely as the load on the bearing, thus making the product ofthe load on the bearing and the coefficient of friction a constant. Mr Tower's experiments were carried out at nearly constant temperature. The more recent experiments of Lasche (Zeitsch, Verein Deutsche Ingen., 1902, 46, 1881) show that the product of the coefficient of friction, the load on the bearing, and the temperature is approximately constant. For further information on this point and on Osborne Reynolds's theory of lubrication see
End of Article: CHAPTER II 

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