MACHINE. 



MACHINE. 



384 



actions of weights,'springs, wind, water, steam, or fired gunpowder ; and 

 these powers may generally be considered as pressures exerted during 

 certain portions of time. Even that power which is produced by a 

 sudden impulse, as when a rammer descending by its weight falls on 

 the head of a pile, is only a pressure existing during an indefinitely 

 short interval of time. The point in any machine to which the moving 

 power is applied is called the impelled, and that against which the 

 resistance acts is called the working point. 



In the employment of any machine a certain portion of the power 

 is expended in overcoming the inertia and friction of the materials, 

 and that which remains is the only efficient force by which the useful 

 effect is to be obtained. Thus, in pushing or drawing a body up an 

 inclined plane, the effective motive power is less than that which is 

 actually expended by as much as is necessary to overcome the inertia 

 of the body and its friction on the plane ; and these might be avoided 

 if it were possible or convenient to raise the body vertically to an 

 equal height by the descent of a body of greater weight, when both 

 are connected together by a string passing over a pulley. The loss 

 of power from inertia is doubled when a reciprocating motion exists 

 in the same machine; for a momentary state of rest takes place 

 between every two contrary directions of the movement, and imme- 

 diately afterwards a .new inertia is to be overcome. The retarding 

 forces above mentioned are evidently greater as the quantity of 

 machinery in an engine is augmented ; and hence every machine should 

 be as simple as possible consistently with the requisite relation between 

 the moving power and the opposing resistance. 



In the construction of machinery it is evident that all abrupt 

 variations of velocity should be prevented, on account of the irregularity 

 which they induce in the action. When, for example, one wheel 

 drives another by means of the teeth on their circumference, the 

 pressure of the teeth takes place wholly on one side of the latter, and 

 the movement may be steady if the teeth are well formed ; but on a 

 sudden diminution of the velocity of the driving-wheel, that which 

 is driven, continuing for a time to move with its actual velocity, 

 tends further to retard the movement of the other, and the pressure 

 of the teeth against each other takes place on the opposite side. Thus 

 a shaking motion is produced which diminishes the efficacy of the 

 machine. The disadvantage attending such variations in the move- 

 ment of the machinery renders it advisable to gaui the required effect 

 by continued pressure, if possible, rather than by the employment of 

 percussive forces. 



It is also a maxim assented to by engineers that the impelled point 

 of a machine should not be allowed to move with a greater velocity 

 than that with which the motive power can act upon it ; since in this 

 cage the excess of velocity in the machine will be employed in accele- 

 rating the motion of the power, and thus the general acceleration of 

 the machine will suffer a corresponding diminution. The velocities of 

 the impelled and working points should therefore be properly adjusted 

 to the pressures, the inertia, and the friction, in order that all possible 

 advantage may be derived from the machine. 



A just estimate of the power of a machine ought to include the 

 effects of all the momentary accelerations and retardations of motion to 

 which it is subject, and all the losses arising from inertia and friction ; 

 but as the introduction of these circumstances would excessively com- 

 plicate the investigation, it is usual to make the measure of the power 

 depend on the condition that the impelled and working points shall be 

 in a state of uniform motion. For then, agreeably to the property of 

 the simple lever, the velocities of those extreme points will be in- 

 versely proportional to the forces which would be in equilibrio at the 

 game points ; and the rule propounded by Euler is, that in every 

 machine, simple or complex, the pressure at the impelled point, 

 multiplied by the velocity of that point, is equal to the product of the 

 resistance at the working point by the velocity of the same point ; or 

 the momentum of resistance (commonly called the performance of the 

 machine) is equal to the momentum of impulse. Whatever objection 

 may be made to this rule with respect to the measure of the power in 

 action, no doubt can exist that it affords a correct value of the useful 

 effect; and the latter may therefore be measured by the weight 

 which might be raised by the machine to a given height vertically in 

 a given time. The fact is 'sufficiently evident when a mass of any 

 material is to be conveyed from one place to another, or when a body 

 is let fall on any object from a given height. It follows that, if an 

 algebraical expression be obtained for the momentum of the resistance 

 in terms involving that resistance, the motive power and the distances 

 of their points of application from the axis of motion, on making the 

 differential of that expression equal to zero, the ratio of the resistance 

 to the moving power, When the useful effect of the machine is a 

 maximum, may be found from the resulting equation. 



If H represent the mass of any body moved, w its weight, which is 

 equal to ug,y ( = : J2J feet) expressing the force of gravity; also, if H 

 be the height to which the body may be raised in one second of time, 

 and v the velocity which a body would acquire by falling vertically 

 through a height equal to H, we shall have, by the theory of motions, 

 V 5 =2</H; whence WH (the momentum of resistance, or the useful 

 effect of a machine) = J M V J . This last expression is designated the 

 /. or active, force of the body moved; and it expresses the force 

 of a body in motion, in contradistinction to the simple pressure exer- 

 cised by a body at rest 



It is commonly asserted that, in the employment of machinery, as 

 much is lost in time as is gained in power, or that the momentum 

 of resistance is proportional to the power employed; but this rule 

 requires some modification. It can be shown to hold good in a well- 

 constructed machine when the object moved resists by its inertia only ; 

 but if the inertia is but a small part of the resistance, the momentum 

 of the latter, or the work done, is found to increase nearly as the square 

 of the power employed. 



The various ingenious contrivances which have been adopted in 

 machines for regulating the velocities, and for converting one species 

 of motion into another, are noticed in the article WHEELS. 



In investigating the relations between the motive powers and resist- 

 ances to be overcome, which render the effect produced a maximum 

 with respect to quantity of motion or velocity, or which render the 

 time of the performance a minimum, it is usual to consider that in 

 every machine there is a certain point at which, if the moving power 

 were immediately applied, and a certain point at which, if the resist- 

 ance to be overcome were immediately applied, the effect produced 

 would be the same as that which is produced by the machine in its 

 actual state. Thus, in a machine consisting of several wheels and 

 axles with which weights are raised by means of ropes passing over 

 their circumferences, the points at which the ropes immediately con- 

 nected with the moving power and resistance are tangents to the cir- 

 cumferences are those at which the forces are .conceived to be applied. 

 Also, if several forces act at once as moving powers, and resistances 

 are to be overcome at once at various points, the resultant of all the 

 forces and that of all the resistances must be taken for the effective 

 moving power and the effective resistance. The points of application 

 of these resultant forces are to be found, and at these points such 

 resultant forces are conceived to be applied : the effects of friction, 

 the rigidity of ropes, and every other impediment to the action of the 

 machine, are also to be estimated and applied as additions to the 

 resistance which is to be overcome ; and thus a complex machine is 

 reduced to an equivalent mechanical power of a simple form. 

 The velocities of the points at which these resultant forces are con- 

 ceived to be applied are equal to the velocities of the power and 

 resistance. 



The motion in machines may be of two kinds. On the application 

 of force to a machine previously at rest a certain movement is induced, 

 and this movement for a time is accelerative ; but in some machines, 

 after a while, the resisting power and the friction of the materials 

 destroy the acceleration, when, unless the machine is subject to varia- 

 tions of force, as is the case with those Which are impelled by the wind 

 or by the force of men or animals, the movement will become uniform. 

 On the other hand, there are machines which are acted on by a con- 

 stantly a?celerative power, as when a weight at one end of a rope pass- 

 ing over a wheel descends from an elevated place and raises a weight 

 attached to the other extremity. 



If the velocities of the points of application of the equivalent forces 

 are uniform, a simple equation will express the dynamical equilibrium 

 of the machine; for, F representing the moving power, and v the 

 velocity with which it moves, / the force of resistance and v its velo- 

 city, we have in the case of equilibrium 



FV=/I; 



the first member of the equation is frequently designated the 

 momentum of impulse, and the second the effect produced by the 

 machine. 



But the effect of a moving power on a machine in motion is different 

 from that of an equal power on a machine at rest ; for the effect pro- 

 duced by any constant power in the former case depends upon its 

 relative velocity, or the difference between its own velocity and that of 

 the machine, and, by dynamics, it varies with the square of the rela- 

 tive velocity. Therefore, in order to introduce the absolute effect of a 

 force into the equation of equilibrum in place of the efficient force. 

 there must be given the velocity which would render the force quite 

 ineffectual, as well as the actual velocity of the point of application : 

 let the former be represented by v', and the latter by v ; than F' repre- 

 senting the absolute force when the velocity is zero, and F the actual 

 force when the velocity is v (F' being determined by the weight or 

 resistance which is just sufficient to prevent the power from communi- 

 cating motion to the machine, and v' by the velocity with which the 

 machine can move when the resistance is zero), 



F' : F : : v' 2 : (v' v) 5 ; 

 whence F = ' 3 



y'3 



Then the first member of the equation F \=fv becomes 



or, putting ir 1 for v' v, which gives v \'v', it becomes 

 F' 



Now, in order to find the velocity which is consistent with the pro- 

 duction of the greatest effect by the machine, this expression, which 

 represents the equivalent oifv, the efficient action of the machine, ia 



