33 



RESISTANCE. 



RESISTANCE. 



be the fraction in question. If, then, Q be called the resistance of the 

 air sought, A = the area of the exposed end, v= the velocity of motion, 

 ( = a co-efficient depending upon the length of the train, and 8 = a 

 constant co-efficient to represent the resistance caused by the intervals, 

 the resistance of the air will be represented by the formula 



It thence follows that the sum of the calculable resistances to be 

 overcome when a tram moves in a straight line, on a level road, may 

 be represented (calling the resistance T) by the formula, 



T =/P- +/'(P+p) + 0e A v-. 



R " 



When the train moves upon an incline forming with the horizon an 

 angle a, the gravity of the train is decomposed into two portions, one 

 of which is perpendicular to the plane of the rails, and the other acts 

 in a direction parallel to that plane, and tends to draw the train down- 

 wards ; so that in pulling a train up an incline, the locomotive must 

 exercise a power, not only able to overcome the friction, but also to 

 overcome this effort of gravity in the direction parallel to the surface 

 of the incline. The latter force, as is well known, is equal to (P+p) 

 gin a ; the pressure of the bearings of the boxes upon the axles is 

 equal to P cos a ; and the pressure of the wheels normally to the rails 

 is equal to (P+p) cos a. The force, then, which the locomotive must 

 exercise upon a waggon in order to allow the latter to retain the 

 velocity it had at any particular moment, under the conditions 

 supposed, must be 



/p cos a I+/'(p +p) cos a + 8tAv"- + (r + p) sin a. 



As the inclines upon railways are generally such that we may consider 

 cos o = l, and that sin a=tga, we may consider that the force exer- 

 cised may, practically, be represented by the formula, 



/P =/'(P+p) + 0fAV*+ (P+P) tga. 



R 



The resistances a waggon encounters on a curve are of a complicated 

 nature, and in a railway waggon they become more than usually so, 

 from the fact that the wheels are fixed on the axles. Calling a the 

 half width of the way, and p the mean radius of the curve, if the centre 

 of the waggon travel over a space = 1, the interior wheels will travel 



? , and the exterior wheels ? ; so that the slip of each of them will 



P P 



be . Now the wheels exercise a pressure = P + p, and if we repre- 



P 

 sent the co-efficient of friction by /", the expression of the resistance 



created by the fixity of the wheels will become /"(P +p) - for every 



unity of distance. But there is another resistance developed on a 

 curve from the two axles of a carriage being fixed to its frame ; for 

 whilst the centre of the carriage traverses a circle described by the 

 radius o (the mean radius of the curve), the respective points of con- 

 tact of the axles describe a circumference around the centre o of the 

 rectangle formed by the points of contact of the wheels and of the 

 rails. The radius corresponding to these points of contact is o A = 

 V z + b', b representing the half distance of the two axles, and " tin- 

 half distance of the two points of contact of tho wheels on the same 

 axle. The slip of the wheels, whilst the waggon makes a revolution 

 round o, becomes then /" (P + p) 2 T y' a* + 4 3 . In the same space of 

 time, the distance traversed by the waggon = 2 wp ; so that the resist- 

 ance caused by the wheels being fixed on their axles, and by the 

 parallelism of those axles themselves, is represented by the formula, 



2 %/' + ** Vo 2 + * s 



= i (p + tt\ 



%*P J I' 



There is a third force developed during the curvilinear movement of 

 a waggon arising from the pressure of the rims of the wheels upon the 



p + p V* 

 outer rail, whose expression is ; in which g represents the 



accelerating force of gravity, which is in our latitude = 32^ feet. This 



p + p v 3 

 pressure gives rise to a friction,/"' , in which /'" represents 



the co-efficient of the friction created by the pressure of the rims of the 

 wheels upon the inner surface of the outer rail. But in addition to 

 this there is a friction exercised by the surface of the rim in its motion 

 along the inner edge of the rail*, and the resistance thus developed by 

 the friction due to the centrifugal force, for every unity of the advance 



p+p V* A/2RA + If 

 of the waggon, is / ; in which n = the radius 



of the inner edge of the rim of the wheel, and A = the depth of that 

 rim. The additional resistance thus created by the passage of a train 

 over a curve in, then, represented by 



P ff P B 



The total effort which the locomotive must exercise in order to retain 

 the velocity it previously had, must, during the unity of the distance 

 ARTS AHD SCI. DIV. VOL. VII. 



traversed, be equal to the sum of the resistances above stated ; and it 

 is therefore represented by the formula 



P B 



In Perdonnet's ' Traite" Elementaire,' in Wood's ' Treatise on Rail- 

 ways,' in the second edition of M. de Pambour's ' Traitd des Machines 

 Locomotives,' &c., the various experiments by which the values of the 

 various co-efficients in the above formute have been ascertained, are 

 discussed at length. It may suffice here to say that generally speaking 

 the values assigned by Coulomb to the co-efficients /,/", are considered 

 still to apply, though they are evidently in excess of the actual values 

 as they have been indicated by experiments on railway trains ; for Mr. 

 Wood found that the total resistance of waggons of the old model 

 travelling, at small velocities, upon a level straight lice, was 0-00475 

 (p + p), and De Pambour found it to be only 0'0026 (e+p). Messrs. 

 Gouin and Lechatellier found, by direct experiment with Morin's 

 dynamometre, that the resistance from this cause was, at speeds vary- 

 ing between 15 to 25 miles per hour, equal to 0'003 to Ci'0045 (P+p) ; 

 at speeds between 25 and 38 miles, it was = 0-0045 to 0-0085 (P+p) ; 

 and for speeds between 50 and 60 miles, it was = 0'012 to O'Olo (P +p) 

 On the average it may be taken that these resistances amount to 

 O'OO t (P+p). 



De Pambour gives the value of 9=0-004823, and that of e = l'17 

 for a cubical body, =1'07 for a train of five carriages, =1'05 for a 

 train of fifteen carriages, and =1'04 for a train of twenty-five carriages ; 

 but the learned author seems to consider the truest value of e to be 

 = 1'13. It is generally considered that the value of /" = 0'16; the 

 value of /'" has not been ascertained by direct experiment, but it is 

 known to be considerably greater than f". 



The practical conclusions drawn from the examination of these for- 

 mulae are: 1. The resistance is diminished by diminishing the diameter 

 of the axles, and increasing that of the wheels. 2. It is desirable to 

 diminish the weight of the carriages as far as may be consistent with 

 safety. 3. The resistance in traversing a curve is increased in propor- 

 tion as the radius is decreased. 4. The useful effect of locomotives is 

 materially affected by their rate of motion, and economically there are 

 serious objections to high rates of speed. 



The experiments of Messrs. Gouin and Chatellier show also that a 

 great portion of the resistance of a train arises from the frictions pro- 

 jjiuced by the frame of the engine, and the frictions of its machinery. 

 They found that, in fact, when the resistance of the whole train was 

 equal to 0'0105 per ton, that of the carriages alone was 0'00025 ; that 

 created by the frictions of the machinery alone, without reference to 

 the load, was 0-0025 ; and the resistance created in the machinery by 

 the pressure of the steam, was 0-00175. The 'speed in this case was 

 about 28 miles per hour, and the experiments were tried on an incline 

 of 1 in 125 ; the weight of the train was 60 tons, and the weather quite 

 calm. Finally, it may be stated that it is usually considered that 

 taking into account both the variable and the constant resistances to 

 be encountered on railways the force required to ensure the traction at 

 speeds of 20 miles an hour is equal to J,-, th of the load ; and that if the 

 speed should exceed 30 miles an hour it becomes ^th of the quantity. 



The method of finding the resistance which an engine opposes to 

 the effort made by the steam to put it in motion, is as follows : 

 Multiply the area of one of the two equal pistons in square inches by 

 the pressure of the steam on a square inch of the piston in each 

 cylinder, when that pressure is just sufficient to cause the engine to 

 move ; the product is the pressure on each piston. Then, since the 

 piston makes two strokes while the wheel -of the engine turns once 

 round, the velocity of the piston is to that of the engine as twice the 

 length of the stroke is to the circumference of the wheel ; and, the 

 resistances being inversely proportional to the velocities, we have 



circumf. of wheel : twice the length of tho stroke : : pressure on 



both pistons : the resistance, or inertia, of the engine. 

 But the resistance increases with the load which the engine has to draw ; 

 and, in order to determine it when attached to a train, the above pro- 

 portion may be used ; but the pressure on the pistons, instead of being 

 found as before, must be taken when the engine and train are observed 

 to have a uniform motion. Then the fourth term of the proportion 

 being diminished by the known resistance of the train, will give the 

 ire of the engine alone. 



From the experiments of Mr. Telford, the following values of the 

 resistances experienced by loaded carriages on level roads have been 

 determined. On a good pavement the resistance is ^ of the weight of 

 the carriage and load ; on a broken surface of old flint, J 3 ; on gravel, jV ; 

 and on a well-constructed railway, from a V> to ^ 



By experiments made on the force (of traction) required to give 

 motion to vessels on canals, it is found that the resistance varies nearly 

 as the cube of the velocity; and this great deviation from the general 

 law of resistances is probably caused by the reaction of the sides of the 

 canal against the water displaced by the vessel. It deserves, however, 

 to be mentioned, that when the velocity of the vessel is considerable, 

 the resistance has been found to experience some diminution, perhaps 

 on account of the water momentarily displaced, from its inability to 

 escape laterally, becoming condensed, and thus giving superior buoyancy 



