Volume 2 — Chapter XXXII.—LEATHER BELTING. (continued 2)
Reading edition
263 160 120 140 20 7 2 .8 5 .6 172 .648
264 200 160 180 20 9 5 .1 10 .2 171 .736
265 230 200 215 15 14 .3 7 .3 14 .6 168 .907 1 m.
266 230 200 215 15 14 .3 7 .9 15 .8 168 .907 5 m.
267 270 240 255 15 17 10 .7 21 .4 168 .966 1 m. Belt slipped off 3 m. later.
268 300 350 240 295 55 5 .36 1 .4 2 .8 177 .544
269 400 320 360 40 9 3 6 175 .719
270 450 400 425 25 17 6 .8 13 .6 172 .943 1 m. Belt slipped off 3 m. later.
271 418 360 389 29 13 .4 8 .8 17 .6 173 .859 1 m.
272 418 360 389 29 13 .4 15 .6 31 .2 173 .859 5 m. Belt slipped off 2 m. later.
273 600 700 560 630 70 9 6 .3 12 .6 177 .711
274 650 480 565 85 6 .65 3 .1 6 .2 178 .610 1 m.
275 650 480 565 85 6 .65 3 .9 7 .8 178 .610 5 m.
276 650 480 565 85 6 .65 4 .4 8 .8 178 .610 10 m.
277 600 652 400 526 126 4 .17 1 .4 2 .8 178 .460 One day later.
279 715 560 637 .5 77 .5 8 .23 2 .4 4 .8 177 .682
280 705 560 632 .5 72 .5 8 .72 2 .8 5 .6 177 .701
281 700 560 560 630 70 9 3 6 177 .711
282 560 750 640 695 55 12 .6 4 .1 8 .2 176 .824 1 m.
283 735 535 640 682 .5 47 .5 14 .3 22 44 176 .866 5 m. Belt slipp’d off.
284 770 640 705 65 10 .7 5 .4 10 .8 177 .767 1 m. After 3 min. intermission.
285 300 350 240 295 55 5 .36 1 .2 2 .4 177 .543 Temp. 52°.
286 400 320 360 40 9 1 .8 3 .6 175 .719
287 430 360 395 35 11 .3 2 .7 5 .4 174 .798
289 465 400 432 .5 32 .5 13 .3 5 .3 10 .6 174 .852
290 455 400 427 .5 27 .5 15 .5 7 .3 14 .6 173 .907
291 460 400 430 30 14 .3 11 .6 23 .2 173 .881
292 100 100 44 72 28 2 .57 .5 1 176 .307
293 125 80 102 .5 22 .5 4 .55 .8 1 .6 173 .502
294 165 120 142 .5 22 .5 6 .33 1 .2 2 .4 173 .611
295 200 160 180 20 9 2 .1 4 .2 171 .736
296 230 200 215 15 14 .3 3 .4 6 .8 168 .907
297 230 200 215 15 14 .3 3 .9 7 .8 168 .907
298 100 270 240 225 15 17 5 .7 11 .4 168 .966 1 m.
299 270 240 255 15 17 7 .6 15 .2 168 .966 5 m. [4 m. later.
300 270 240 255 15 17 9 .3 18 .6 168 .966 10 m. Belt slipped off
303 100 110 40 75 35 2 .14 .1 .4 177 .246 20 in. pulleys.
304 132 80 106 26 4 .08 .4 1 .6 174 .463
305 160 120 140 20 7 1 4 172 .648
306 195 160 177 .5 17 .5 10 .1 1 .9 7 .6 169 .814
307 230 200 215 15 14 .3 3 12 168 .907 1 m.
308 230 90 200 215 15 14 .3 3 .5 14 168 .907 5 m.
309 270 240 255 15 17 4 .5 18 168 .966 1 m.
310 270 240 255 15 17 5 .8 23 .2 168 .966 5 m.
311 270 240 255 15 17 6 .2 24 .8 168 .966 10 m.
312 270 240 255 15 17 6 .2 24 .8 168 .966 15 m. Temp. 56°.
313 270 240 255 15 17 2 8 168 .966 1 m.
314 270 240 255 15 17 2 .1 8 .4 168 .966 5 m. Temp. 42°.
315 305 280 292 .5 12 .5 23 .4 3 .4 13 .6 165 1 .09 1 m.
316 305 100 280 292 .5 12 .5 23 .4 3 .5 14 165 1 .09 5 m.
317 100 335 320 327 .5 7 .5 43 .7 5 .2 20 .8 152 1 .42 1 m.
318 335 320 327 .5 7 .5 43 .7 6 .5 26 152 1 .42 5 m.
319 300 380 320 350 30 11 .7 1 .3 5 .2 173 .814 1 m.
320 380 320 350 30 11 .7 1 .4 5 .6 173 .814 5 m.
321 440 400 420 20 21 2 .1 8 .4 170 1 .03 1 m.
322 440 260 400 420 20 21 2 .4 9 .6 170 1 .03 5 m.
323 300 480 440 460 20 23 2 .8 11 .2 170 1 .06 1 m. Temp. 46°.
324 480 285 440 460 20 23 3 12 170 1 .06 5 m.
325 510 480 495 15 33 3 .2 12 .8 167 1 .20 1 m. [lat. Pul. warm.
326 510 480 495 15 33 5 20 167 1 .20 5 m. Belt sl. off 5 m.
A peculiar and important feature of Tables III. and IV. is the effect of time upon the percentage of slip. In previous experiments the percentage of slip was measured at once after the load was applied, but it was accidentally discovered that repeated measurements seldom agreed, and investigation showed that these discrepancies were principally due to the duration of the experiment. The continual slipping of the belt was found to cause a deposit of a thick black substance upon the surface of the pulley, which, acting as a lubricant, continued to increase the slip still further.
Upon removing the load on brake-wheel, this deposit would be again absorbed by the belt, and the original adhesion would be restored. The temperature was also found to affect the slipping, and, in general, the colder the weather the slower would this deposit take place.
Experiments 353 to 360 inclusive were made to determine the limit at which the belt would run continuously without increasing its percentage of slip. After the pulleys had become well coated and the slip had reached a high per cent., the load on the brake-wheel was gradually removed until a marked improvement was reached, as shown by experiments 359 and 360. The highest allowable coefficient of friction for this belt is therefore estimated to be somewhere between 1.13 and .995, or we may safely say 1. The highest coefficient obtained was 1.67, but, of course, this was temporary. The diameter of the pulley also appears to affect the coefficient of friction to some extent. This is especially to be noticed at the very slow speed of 18 revolutions per minute on 10 in. and 20 in. pulleys, where the adhesion on the 20 in. pulleys is decidedly greater; but, on the other hand, at 160 revolutions per minute the adhesion on the 10 in. pulleys is often as good, and sometimes better, than appears for the 20 in. at the same velocity of sliding.
It might be possible to determine the effect of pulley diameter upon adhesion for a perfectly dry belt, where the condition of its surface remains uniform, but for belts as ordinarily used it would be very difficult, on account of the ever-changing condition of surface produced by slip and temperature. It is generally admitted that the larger the diameter the greater the adhesion for any given tension, but no definite relation has ever been established, nor, indeed, does it seem possible to do so except by the most elaborate and extensive experiments.
It should be observed, however, that such a variation, if true, implies a corresponding variation in the coefficients of friction for different intensities of pressure upon the same pulleys, and that, consequently, our experiments should show higher coefficients under the lighter loads for the same velocity of sliding. Referring to Table II., where the condition of the belt is dry and uniform for a large range of tensions, we find that this inference is generally sustained, although there are some few exceptions.
Experiment 106 may be compared with 116, and 112 with 133, also 108, 113, and 135, all showing great reductions in the coefficients of friction for increments in tension. The exceptions are all to be found under the smallest velocities of sliding, and appear only in the third decimal place, so that the weight of their record against the probability of such a law is light. By a similar inference it should also follow that a wide belt would drive a little more at a given tension than a narrow one, on account of the reduction in pressure per square inch against the pulley. The mean intensity of pressure of a belt against its pulley may be considered as proportional to the sum of the tensions divided by the product of pulley diameter and width of belt, and an analysis of the experiments referred to will show the relation there existing between intensity of pressure and coefficient of friction.
If we let Ι = intensity of pressure, and φ = coefficient of friction, we shall find that φ is approximately proportional to Ι-.15, or, in other words that doubling the width of belt or diameter of pulley would apparently increase the coefficient of friction about 10 per cent. of its original value. This relation is not proved, of course, and it is given only as a suggestion toward the solution of the question. If the coefficient of friction does vary with the intensity of pressure, the problem of determining the driving power of a belt on strictly mathematical principles will indeed be complicated.
The coefficient of friction in the tables has been calculated by a well-known formula, developed upon the assumption of a uniform coefficient around the arc of contact, but this could no longer be considered as correct if the coefficient is known to vary with the pressure. Referring from Table II. to Table III., we shall find at once the proof and contradiction of the inferences drawn from Table II., and we are left as much in the dark as ever respecting the value of pressure intensity.
Practical millwrights all know, or think they know, that an increase of pulley diameter increases the drive, and it is a matter of common observation that when large and small pulleys are connected by a crossed belt, the smaller pulley will invariably slip first.
On one side a great deal of testimony can be adduced to show that pressure intensity should be an important factor in the theory of belt transmission, and, on the other hand, we have strong evidence to the contrary. I may refer, in this connection, to the experiments of Mr. Holman in Journal of Franklin Institute for September, 1885, in which there is no indication that the coefficient of friction varies at all with the pressure. The coefficients obtained by Mr. Holman follow the variations in slip like our own, and it gives us pleasure to observe that our general results and conclusions are so strongly corroborative of each other. There is at the same time a great difference in the methods pursued in arriving at the same results. In his experiments, the velocity of sliding was the fixed condition upon which the coefficient of friction was determined, while, in ours, the conditions were those of actual practice in which the percentage of slip was measured. Our least amount of slip, with a dry belt running at the extremely slow speed of 90 feet per minute, was 1.08 inches, and ten times this would be perfectly proper and allowable. A great many of Mr. Holman’s experiments are taken at rates below 1′′ per minute, and the coefficients obtained are very much below the average practice, as himself seems to believe.
The velocity of sliding which may be assumed in selecting a proper coefficient is directly proportional to the belt speed, and may safely be estimated at .01 of that speed. For a pair of pulleys we should have .01 on each pulley, and therefore .02 for slip. Few belts run slower than 200 or 300 ft. per minute, and consequently a slip of less than 2 or 3 ft. per minute need seldom be considered. Another point of difference which may possibly affect the coefficients obtained, is that, in Mr. Holman’s case the same portion of belt surface was subject to continuous friction, while in ours, the friction was spread over the belt at successive portions as in actual work. This we consider a new and important feature of our experiments. As a matter of practical importance, care was taken to observe, as nearly as possible, the maximum slip which might safely take place before a belt would be thrown from its pulley. A number of observations taken throughout the experiments led to the final conclusion that 20 per cent. of slip was as much as could safely be admitted. This information has been found of value in cases where work is done intermittently by a fly-wheel and the belt has to restore the speed of the wheel. It cannot be said in regard to a maximum value of [phi] that any was determined or even indicated, although it is certain that the increase at high rates of slip becomes less rapid.
We have now seen that the driving power of a leather belt depends upon such a variety of conditions, that it would be manifestly impracticable if not impossible to correlate them all, and it is thought better to admit the difficulties at once than to involve the subject in a labyrinth of formulæ which life is too short to solve.
The relative value of pulley diameters may vary with different belts, and all that can be expected or desired is some general expression covering roughly the greatest number of cases. Our apparatus did not admit of extensive variations in this respect, and our attention was given principally to the question of slip.
The coefficients given in Table III. are remarkably high, and show a great superiority for the rawhide over tanned leather in point of adhesion. The belt in question was very soft and pliable, but a little twisted from use on a cone pulley where it had rubbed against one side. It is not desirable, on account of its soft and adhesive nature, to use this kind of belt where frequent shifting is required, and when used on cone pulleys it is liable to climb and stretch against the side of the cone; but for a plain straight connection, there seems to be little room for improvement. Table IV. contains the results of similar experiments upon an oak-tanned leather belt made by Chas. A. Shieren & Co. Here the coefficients are much smaller than those given in Table III., and there is quite a marked difference between the coefficients for 10 in. and 20 in. pulleys.
As before noticed, the outside temperature has its effect, and it is probable that much lower results would have been obtained had the experiments been made in the heat of midsummer. The high coefficients obtained, together with the rapid increase of tension, show that the pulling power of a long horizontal belt must, in many cases, be limited by its strength rather than by its adhesion.
Table V. gives the results of experiments upon a light planer belt at very slow and very high speeds. As would naturally be expected, much higher coefficients were found at the high speed on account of the greater velocity of sliding.
TABLE V.
OAK-TANNED LEATHER BELT 2′′ WIDE BY 3⁄16′′ THICK AND 30′ 4′′ LONG, WEIGHING 4 LBS., ON 20′′ CAST-IRON PULLEYS. DRY AND SMOOTH, TAKEN FROM SERVICE ON PLANER.
No. of Exper- i’nt. Sum of Tensions T + t T - t Work- ing. T t T ⁄ t Per- cent- age of Slip. Velocity of Slip in ft. per min. Arc of con- tact. Coef- ficient of Friction. Dura- tion of run at time of experi- ment. Remarks.
Initial. Work- ing. Final.
429 100 110 40 75 35 2 .14 1 .2 .54 179 ° .243 18 r.p.m.
430 115 60 87 .5 27 .5 3 .18 6 .1 2 .75 178 .372
431 118 70 94 24 3 .92 16 .5 7 .42 178 .440
432 105 20 62 .5 42 .5 1 .47 .3 .14 179 .123
433 112 50 81 31 2 .61 3 .5 1 .57 178 .309
435 200 204 40 132 82 1 .61 .2 .09 180 .152
436 206 60 133 73 1 .82 .7 .32 180 .191
437 208 80 144 64 2 .25 1 .8 .81 179 .260
438 210 100 155 55 2 .82 3 .7 1 .66 179 .332
439 212 120 166 40 3 .61 7 .7 3 .47 179 .411
440 215 140 177 .5 37 .5 4 .73 18 .4 8 .28 179 .497
442 100 110 60 85 25 3 .40 .3 7 .12 178 .394 950 r.p.m.
443 120 80 100 20 5 .7 16 .62 178 .518
445 125 90 107 .5 17 .5 6 .14 3 71 .25 177 .587 Start.
446 125 90 107 .5 17 .5 6 .14 25 593 .7 177 .587 min.
448 200 200 80 140 60 2 .33 .4 9 .5 179 .271
449 200 100 150 50 3 .5 11 .87 179 .352
450 195 175 120 157 .5 37 .5 4 .20 .8 19 179 .459
451 150 175 120 147 .5 27 .5 5 .36 .9 21 .38 178 .540
452 135 160 120 140 20 7 20 475 178 .626
It may here be mentioned that the sum of the tensions was the horizontal pressure of the belt against the pulleys, and that no allowance was necessary for the effect of the centrifugal force. At the speed here used, the tension indicated in the belt at rest was about 50 lbs. greater than when in motion.
TABLE VI.
SHOWING THE AVERAGE COEFFICIENT OF FRICTION AND VELOCITY OF SLIP FOR A NUMBER OF EXPERIMENTS IN WHICH THE SLIP APPROXIMATED 2 PER CENT.
No. ex- per’ts in av’ge. Per- cent age of Slip. Veloc. of Sl. in ft. per m. Coeffi- cient of Fric- tion. Belt. Pulleys. Remarks.
3 1 .4 5 .6 .661 51⁄2′′ old belt. Table I 20′′ diam. pap. cov’d Belt in nor. w’k’g con.
2 1 .7 6 .8 .44 51⁄2′′ old belt. „ I 20′′ di. cast-iron sur. „ „ „
2 1 .55 6 .2 .575 51⁄2′′ old belt. „ I 20′′ di. cast-iron sur. Belt dressed with “Beltiline.”
5 1 .7 6 .8 .452 21⁄4′′ dbl. belt. „ II 20′′ di. cast-iron sur. B’t dry as us. on plan’r.
2 1 .5 6 .818 21⁄4′′ dbl. belt. „ II 20′′ di. cast-iron sur. Belt dressed with “Sankey’s Life of Leather.”
2 1 .7 6 .8 1 .38 4′′ r’hide b. „ III 20′′ di. cast-iron sur. Belt in nor. w’k’g con.
11 1 .8 3 .6 .861 4′′ r’hide b. „ III 10′′ diameter. „ „ „
1 2 .45 .432 4′′ r’hide b. „ III 10′′ diameter. „ „ „
1 1 .9 .86 .691 4′′ r’hide b. „ III 20′′ diameter. „ „ „
7 1 .94 3 .88 .617 4′′ o.tan’d b. „ IV 10′′ diameter. „ „ „
4 1 .85 7 .40 .906 4′′ o.tan’d b. „ IV 20′′ diameter. „ „ „
2 1 .5 .67 .251 2′′ o.tan’d b. „ V 20′′ diameter. B’t dry as us. on plan’r.
2 .8 38 .529 2′′ o.tan’d b. „ V 20′′ diameter. „ „ „
The conclusion to be drawn from this series of experiments is the great importance of high speed in the economy of belt transmission. The friction of belts on pulleys is evidently dependent on the velocity of sliding, and, as a general rule, the greater the velocity the greater the friction. There are but few apparent exceptions to this rule, and investigation of them has led to the inference that in all such cases, the condition of the belt or pulley surface had undergone a change either by heating or by deposit from the belt on the pulley. The percentage of slip is the measure of the power lost in transmission by the belt itself, and the higher the speed the less this becomes. There is a limit, however, to the power which may be transmitted as the speed is increased, and this limit is caused by the reduction in pressure against the pulley arising from the action of centrifugal force.
This point has been clearly demonstrated in a paper read before this Society by Mr. A. F. Nagle on the “Horse Power of Leather belts,”[43] and the formula there developed is written thus:
HP = CVtw(S - .012 V2) ÷ 550, (1.)
in which C is a constant to be determined from the arc of contact and coefficient of friction as expressed in the equation:
C = 1 - 10-.00758fα, (2.)
V = velocity of belt in feet per second.
t = thickness of the belt in inches.
w = width „ „
S = working strength of leather in lbs. per square inch.
f = coefficient of friction.
α = arc of contact in degrees.
The velocity at which the maximum amount of power can be transmitted by any given belt is independent of its arc of contact and coefficient of friction, and depends only upon the working strength of the material and its specific gravity.
From equation (1.) we obtain for the maximum power of leather belts the condition:
V = √(28S), (3.)
and for any other material whose specific gravity is y, we find
V = 5 × √S/y, (4.)
The coefficient of friction .40, adopted by Mr. Nagle, appears from these experiments to be on the safe side for all working requirements, except in cases where dry belts are run at slow speeds.
If we assume 2 per cent. as the greatest allowable slip, and select within this limit the coefficient corresponding to the nearest approximations to it, we can form some idea of the coefficients which can be relied upon at different speeds.
Table VI. gives the average results obtained for this maximum allowance of slip, and shows an extreme variation in the coefficient of friction from .251 for a dry oak-tanned belt at the slow speed of 90 feet per minute to 1.38 for a rawhide belt at the moderate speed of 800 feet per minute.
For continuous working, it is probable that the coefficient 1.38 is too high, but still it is certain that a coefficient of 1.00 can be steadily maintained for an indefinite length of time, and we may say that in actual practice the coefficient of friction may vary from .25 to 1.00 under good working conditions. This extreme variation in the coefficient of friction does not give rise, as might at first be supposed, to such a great difference in the transmission of power. It will be seen by reference to formula (1.) that the power transmitted for any given working strength and speed is limited only by the value of C, which depends upon the arc of contact and the coefficient of friction.
For the usual arc of contact, 180°, the power transmitted when f = .25 is about 24 per cent. less than when f = .40, and when f = 1.00, the power transmitted is about 33 per cent. more, from which it appears that in extreme cases the power transmitted may be 1⁄4 less or 1⁄3 more than will be found from the use of Mr. Nagle’s coefficient of .40.
TABLE VII.
SHOWING THE TORSIONAL MOMENT IN LBS. REQUIRED TO OVERCOME JOURNAL FRICTION AND OTHER INTERNAL RESISTANCES, FOR BELTS AT VARIOUS SPEEDS AND TENSIONS ON DIFFERENT ARRANGEMENTS OF PULLEYS.
No. of experi- m’nt. Ten- sion. T + t Moment in inch lbs. Dia- meter of pul- leys. Revo- lut’s per min. Width of Belt. Thick- ness of Belt. Manner of Driving. Remarks.
1 100 20 20′′ 160 6′′ 7⁄32′′ Straight open belt.
3 300 25
5 500 30
7 700 35
10 1000 45
45 100 15
47 300 22 .5
49 500 27 .5
51 700 35
54 1000 50
163 100 17 .5 20′′ 160 4′′ 9⁄32′′ Straight open belt.
165 300 25
167 500 30
169 700 35
194 100 17 .5 10′′ 160 4′′ 5⁄16′′ Straight open belt.
196 300 27 .5
198 500 40
200 700 55
202 900 70
203 1000 80
327 100 20 10′′ 18 4′′ 5⁄16′′ Straight open belt.
328 1000 80
393 100 20
394 1000 100
395 600 60
405 100 20 20′′ 18 4′′ 9⁄32′′ Straight open belt.
406 1000 160
407 600 100
428 100 20 20′′ 18 2′′ 9⁄32′′ Straight open belt.
434 200 25
441 100 25 20′′ 950 2′′ 3⁄16′′ Straight open belt.
447 200 30
453 100 25 20′′ 160 6′′ 7⁄32′′ Crossed belt. 14′ 6′′ between pulleys.
454 500 60
455 1000 110 14′ 6′′ bet. pul’ys.
459 100 15 20′′ 160 6′′ 7⁄32′′ Straight open belt. 14′ 6′′ between pulleys.
460 500 25
461 1000 65
462 100 25 20′′ 160 6′′ 7⁄32′′ Straight open belt. With 8′′ tightener.
463 500 60
464 1000 110
465 100 45 20′′ 160 6′′ 7⁄32′′ Crossed belt. 8 feet between pulleys.
466 500 105
467 1000 180
470 100 25 20′′ 160 6′′ 7⁄32′′ Quarter turn belt on 16′′ diameter mule pulleys.
471 500 80
472 750 145
473 1000 250
474 750 170
475 500 110
476 1000 220
477 1000 140 20′′ 160 6′′ 7⁄32′′ Quarter turn belt on 16′′ diameter mule pulleys. Freshly oiled.
478 750 100
479 500 70
480 100 20
481 50 60 20′′ 160 6′′ 7⁄32′′ Quarter turn on 16′′ mule pulleys. Belt rub. against low. guide m. pul.
482 25 120
483 100 20 20′′ 160 6′′ 7⁄32′′ Quarter turn on 16′′ mule pulleys. Well oiled, after a run of 2 hrs. at T + t = 100.
484 500 50
485 750 70
486 1000 105
495 250 30 20′′ 160 6′′ 7⁄32′′ Half turn belt on 16′′ mule pulleys.
496 500 50
497 750 90
498 1000 170
503 1000 260 20′′ 160 6′′ 7⁄32′′ Quarter twist. 10 feet between pulleys.
504 750 190
505 500 130
506 250 80
507 100 30
513 100 50 20′′ 160 6′′ 7⁄32′′ Quarter twist. 7′ 6′′ between pulleys.
514 250 105
515 500 200
516 750 290
517 1000 380
523 100 25 20′′ 160 4′′ 1⁄4′′ Quarter twist. 10 feet between pulleys.
524 250 50
525 500 95
526 750 145
527 1000 210
528 100 65 20′′ 160 4′′ 1⁄4′′ Quarter twist. 6 feet between pulleys.
529 250 135
530 500 245
531 750 380
533 100 25 20′′ 160 6′′ 7⁄32′′ Quarter twist. 16′ 6′′ between pulleys.
534 250 40
535 500 75
536 750 105
537 1000 165
539 1000 130 20′′ 160 6′′ 7⁄32′′ Quarter twist with 16′′ diameter carrying pulley. 7′6′′ between pulleys.
540 750 110
541 500 90
542 250 60
543 100 40
544 100 30
545 250 55
546 500 90
547 750 120
548 1000 170
569 100 25 20′′ 160 6′′ 7⁄32′′ Straight open belt.
571 500 55
572 750 70
573 1000 90
The percentage of slip is the most important factor affecting the efficiency of belt transmission, but in addition to this we have journal friction, the resistance of the air, and with crossed belts the friction of the belt upon itself. These have been termed internal resistances, and their values for some of the most common arrangements of pulleys are given in Table VII. From this table it appears that the moment required to run a straight belt varies from 15 to 25 inch lbs. at 100 lbs. tension for all speeds. At 160 revolutions per minute and 1,000 lbs. tension, the required moment varied from 45 to 90 inch lbs., and at 18 revolutions per minute and at the same tension it varied from 80 to 150 inch lbs.
From the average of these quantities we find the moment of resistance to be expressed by the following formulæ for straight open belts between 2′′ journals:
At 160 r. p. m.:
M = .053 S + 14.7, (5.)
At 18 r. p. m.:
M = .11 S + 9, (6.)
in which
M = moment of resistance in inch lbs.
S = sum of tensions.
When a crossed belt does not rub upon itself, the resistance is the same as for an open belt.
The resistance offered by the introduction of carrying pulleys and tighteners is appreciable, and depends upon the pressure brought to bear against their journals. If the belt rubs against the flanges of the carrying pulleys, the resistance is very much increased, and this is often liable to occur in horizontal belts from a change of load. The friction on journals of carrying pulleys may be estimated by the formulæ already given if we substitute for S the pressure against their journals. In the experiments which were made upon internal resistances, the greatest resistance was offered by a quarter-twist belt 6 feet between journals on 20-inch pulleys.
The equation for this belt may be written:
M = .35 S + 58, (7.)
but the introduction of a carrying pulley reduced the resistance to no more than what might be expected from the same number of journals with a straight belt.
With quarter-twist belts the resistance lies chiefly in slip, which occurs as the belt leaves the pulleys, and this naturally depends upon the distance between journals in terms of the diameters of the pulleys.
The effect of time upon the tension of the belt used in Table VIII. is plainly shown by experiments 588 to 613 inclusive, between which the pulleys remained at a fixed distance apart, and the belt slowly stretched from a tension of 380 to 280 lbs.
To estimate the efficiency of belt transmission for an average case, we may assume 40 in. lbs. as the moment of internal resistance for a belt whose tension is 500 lbs. and 40 in. lbs. statical moment = about 20 ft. lbs. per revolution. If the belt is transmitting 400 lbs. with two per cent. of slip on 20 in. pulleys, then .02 × 400 × 5 = 40 ft lbs. are lost per revolution in slip, making a total loss of 60 ft. lbs. per revolution.
TABLE VIII.
SHOWING THE INCREASE IN THE SUM OF THE TENSIONS ON A VERTICAL BELT 4′′ WIDE BY 1⁄4′′ THICK, AND 24 FT. LONG, ON 20′′ CAST-IRON PULLEYS, AT 120 R. P. M.
No. of experi- m’nt. Scales A. [44] Scales B. [44] Ten- sion T + t Ten- sion T - t T t Incre- m’nt of T + t Per- cen’e of Incre- ment. Date.
578 93 101 194 16 105 89 0 5-15-1885.
579 70 142 212 144 178 34 18
580 67 170 237 206 221 .5 15 .5 43
581 66 180 246 228 237 9 52
582 66 188 254 244 249 5 60 .323
583 91 101 192 20 106 86 -2
584 202 210 412 16 214 214 0 5-15-1885.
585 167 250 417 166 292 .5 292 .5 5
586 145 300 445 310 376 .5 376 .5 33 .171
587 185 195 380 20 200 200 -32
588 190 199 380 0 190 190 0 5-18-1885.
589 133 250 393 214 303 .5 89 .5 13 .033
590 177 177 354 0 177 177 0 5-19-1885.
591 156 203 359 94 226 .5 132 .5 5
592 138 235 373 194 283 .5 89 .5 19
593 135 250 385 230 307 .5 77 .5 31
594 128 275 403 294 348 .5 34 .5 49
595 125 300 425 350 387 .5 37 .5 71
596 123 325 448 404 426 22 94 .333
597 168 168 336 0 168 168 -18
598 143 143 286 0 143 143 0 5-25-1885.
599 140 148 288 16 152 136 2
600 130 160 290 60 175 115 4
601 122 170 292 196 194 98 6
602 116 180 296 28 212 84 10
603 112 190 302 156 229 73 16
604 108 200 308 184 246 62 22
605 105 210 315 210 262 .5 52 .5 29
606 102 220 322 236 279 43 36
607 100 230 330 260 295 35 44
608 99 240 339 282 310 .5 28 .5 53
609 98 250 348 304 326 22 62
610 98 260 358 316 337 21 72
611 99 270 369 342 355 .5 13 .5 83
612 100 280 380 360 370 10 94 .357
613 140 140 280 0 140 140 -6
The total power expended per revolution is about 2,000 ft. lbs., therefore .03 is lost.
Under light loads, the internal resistance, which is nearly constant in amount, may be a large percentage of the power transmitted, while under heavy loads the percentage of slip may become the principal loss.
It would be difficult to work out, or even to use, a general expression for the efficiency of belt transmission, but, from the foregoing, it would seem safe to assume that 97 per cent. can be obtained under good working conditions.
When a belt is too tight, there is a constant waste in journal friction, and when too loose, there may be a much greater loss in efficiency from slip. The allowance recommended of 2 per cent. for slip is rather more than experiment would indicate for any possible crawl or creep due to the elasticity of the belt, but in connection with this, there is probably always more or less actual slip, and we are inclined to think that in most cases this allowance may be divided into equal parts representing creep and slip proper. Under good working conditions, a belt is probably stretched about 1 per cent. on the tight side, which naturally gives 1 per cent. of creep, and to this we have added another per cent. for actual slip in fixing the limit proposed.
The indications and conclusions to be drawn from these experiments are:
1. That the coefficient of friction may vary under practical working conditions from 25 per cent. to 100 per cent.
2. That its value depends upon the nature and condition of the leather, the velocity of sliding, temperature, and pressure.
3. That an excessive amount of slip has a tendency to become greater and greater, until the belt finally leaves the pulley.
4. That a belt will seldom remain upon a pulley when the slip exceeds 20 per cent.
5. That excessive slipping dries out the leather and leads toward the condition of minimum adhesion.
6. That rawhide has much greater adhesion than tanned leather, giving a coefficient of 100 per cent. at the moderate slip of 5 ft. per minute.
7. That a velocity of sliding equal to .01 of the belt speed is not excessive.
8. That the coefficients in general use are rather below the average results obtained.
9. That when suddenly forced to slip, the coefficient of friction becomes momentarily very high, but that it gradually decreases as the slip continues.
10. That the sum of the tensions is not constant, but increases with the load to the maximum extent of about 33 per cent. with vertical belts.
11. That, with horizontal belts, the sum of the tensions may increase indefinitely as far as the breaking strength of the belt.
12. That the economy of belt transmission depends principally upon journal friction and slip.
13. That it is important on this account to make the belt speed as high as possible within the limits of 5,000 or 6,000 ft. per minute.
14. That quarter-twist belts should be avoided.
15. That it is preferable in all cases, from considerations of economy in wear on belt and power consumed, to use an intermediate guide pulley, so placed that the belt may be run in either direction.
16. That the introduction of guide and carrying pulleys adds to the internal resistances an amount proportional to the friction of their journals.
17. That there is still need of more light on the subject.