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Belting facts and figures
.Beltiny Facts and Figures.
317
balanced, the table may be readily moved by one man. tie is provided with an iron bar, one end of which fits to the bolt previously spoken of. He walks along the curb, pushing by means of this bar, and when he arrives at the desired track, immediately drives the bolt home, securing the table where wanted.
BELTING FACTSAND FIGURES, BY J. H. COOPER. (Continued from page 254.)
MR. F. W. B.~coN has kindly furnished me with the following: A certain 13-inch pulley un a shaft running 203 revolutions per minute carries a belt 2"25 inches wide, and drives a 20-inch pulley on a shaft 20 feet vertically above. The pulleys are smooth turned iron and the belt of single leather, with grain side to pulleys. This belt had been running a year or more, under a tension which was limited only by the strength of the lacing. It was used to convey two horse.power to the upper shaft, but was considered by the lessee to be unequal to the task, even when tightly drawn, and its adhesion increased by free application of rosin. It was admitted by both parties that the belt was worked to its fullest capacity. In order to ascertain the exact amount of work done by the belt, the following experiment was made : On the driven shaft above was a smooth turned iron pulley, 6"25 feet in circumference and 4-inch face ; over this was thrown a 3-inch leather belt, with grain side to pulley, and to its ends were attached unequal weights, such that t h e 2"25-inch belt was subjected to its maximum working power. These weights were 203 pounds and 2"25 pounds, Speed of friction pulley was taken at 132 revolutions per minute. Then we have 132 × 6"25 ~ 825 =_ velocity of 3-inch belt in feet per minute, and 825 X 200"75 5"018 ~ horse power of 3-inch ---33,-00b-- = belt. This is equivalent to a driving power of 41"1 square feet of belt per minute per horse.power. Rule for Horse-power of a Belt, from " Overman's Mechanics," p. 414. --One example will explain the table. It is assumed that belts
318
Civil and .~Tecltanica~ Engineering.
should have a speed of 25 to 30 feet per second, and that a 12-inch belt on a 3-feet pulley will transmit 10 horse-power. At 25 feet per second this means a surface velocity of 150 square feet per minute per horse power, which is a liberal allowance. The ~uarter turn belt applied to driving mill stones or upright sl~afts.--A is the driving pulley on horizontal shaft. B the driven pulley on mill spindle or upright shaft, c the tightener, which is placed at the proper angle Ibr receiving the belt from B and delivering it to A ; it has a short shaft running in bearings secured to a
frame which slides vertical!y in fixed grooves, and may be raised to tighten the belt for driving, or lowered to slacken the belt for stopping B, at pleasure. B is made wide and straight on the face to admit of motion in raising and lowering the stones, as well as to allow of lead of belt by the different positions of c, which are due to length and tightness of belt. A and o should be rounding on their faces. The cut shows the proper positions of the pulleys and shafts, and also gives good
working proportions, the particular's having been obtained from machinery in use. W e make a few extracts from a paper " O n the centrifugal force of bands in machinery," b y W. J. M. Rankine, in the Engineer for March 5, 1869, p. 165 : " I t is well known, through practical experience, that a belt for communicating motion between two pulleys requires a greater ten-
BeSting Facts and i'~gures.
3 !9
sion to prevent it from slipping when it runs at a high, than at a low speed. "Various suppositions have been made to account for this, such as that of the adhesion to the belt of a layer of air, which at a very high speed has not time to escape from between the belt and the pulley. But the real cause is simply the centrifugal force of the belt, which acts against its tension, and therefore slackens its grip of the pulleys. " ~'~ It can be proved from the elementary laws of dynamics, that if an endless band, of any figure whatsoever, runs at a given speed, the centrifugal force produces an uniform tension at each cross section of the band, equal to the weight of a piece of the band, whose length is twice the height from which a heavy body must fall, in order to acquire the velocity of the band. " In symbols, let w be the weight of a unit of length of the band ; v the speed at which it runs, and g the velocity produced by gravity in a second ( ~ 32"2 feet) ; then the centrifugal tension (as it may be ~UV 2
called) has the following value :-
g. " T h e effect on the band when in motion is, that at any given point, the tension which produces pressure and friction on the pulleys, or available tension (as it is called), is less than the total tension b y an amount equal to the centrifugal tension ; for this amount is employed in compelling the particles of the band to circulate in a closed or endless path. It is, of course, to the total tension that the strength of the band is to be adapted, therefore the transverse dimensions of a band for transmitting a given force must be greater for a high than for a low speed. "One of the most convenient ways of expressing the size of a band is by stating its weight per unit of length; for example, in pounds per running foot or in kilogrammes per metre. When the size is expressed thus, the corresponding way of expressing the intensity of any stress on the band is in lineal units of itself, such as feet or metres. Let b denote the greatest safe working tension on a band of a given kind, in units of its own length; so that w l is the amount of the safe working tension in units of weight. Let T be the amount of the available tension required at the driving side of the band for the transmission of' power, being usually from two to two and a half times the force to be transmitted. Then the total tension is w v~ T+ =wl. J
320
Civil and Mechanical Engineering.
Whence it is obvious that the required weight per unit of length is given by the following formu]a: T qAT--
V2.
g " F o r example, suppose that the band is a wire rope, that the greatest working tension is to be equivalent to the weight of 2900 feet of the rope, and that it is to run at 100 feet per second: then we have 1 ~ 2900 feet ; V2
--- ~ 310 feet ; g And consequently the weight per running foot of the rope required is : - T ~: ----- 2900--310 ~ 2590; Or about one-eighth part heavier than the rope required, for a speed so moderate as to make the centrifugal tension unimportant. " I n fixing the value of the greatest working tension on a wire rope, a proper deduction must of course be made for the stress produced by the bending of the wires round the pulleys. That stress is given in equivalent length of rope by the expression Ld iij-' where D is the diameter of the smallest pulley round which the rope passes, d the diameter of the wire of which the rope is made, and T. the modulus of elasticity of the wire, in length of itself, viz : about 8,000,000 feet, or 2,400,000 metres. That is to say, let 1 be length of the rope equivalent to the greatest safe working tension on a straight rope; 1t as before, the length equivalent to the actual greatest working tension, then l~d
1_~- ZL---D
" In the case of leathern belts, b may be estimated at about 660 feet, or 200 metres. 't In the case of a leather belt running at the rate of 100 feet per second, the weight per unit of length required, in order to exert a 660 66O given available tension, is increased in the ratio of 660--310 350 or to nearly double, as compared with that of a belt whose centrifugal force is unimportant. " T h e section,~l area of a leathern belt may be calculated approximately in square inches by multiplying the weight per running
A lien En2;rv?.
321
foot by 2"3 ; or in square millimetres, by multiplying tile weight in kilogrammes to the running metre b)~ 1000. " The ordinary thickness of a single belt being about 0"16 inch, or 4 millimetres, tile breadth may be deduced from the sectional area by dividing by that thickness. " The length (L) equivalent to the modulus of elasticity of a leathern belt, as calculated from Bevan's experiments, is about 23,000 feet, or 7000 metres. (To be Continued)
THE ALLEN ENGINE, By CI~X~LES T. PORTI~I~. (Continued from page 252.) IX an engine of 16 inches diameter of cylinder by 42 inches stroke, making 60 revolutions per minute~ and in which the reciprocating parts weigh 600 pounds, the force required to impart and arrest their motion is, at the centres, only 6 pounds on the square inch of piston, diminishing to nothing at the mid-stroke. I f the speed of piston ks increased from 420 feet, as above, to 600 feet per minute, by employing a stroke of 5 feet, and the weight of the reciprocating parts is increased to 1200 pounds, even then the force so absorbed and given out at the centres by these parts is only 18"~ pounds on the square inch of piston. In an Allen Engine, however, of 16 inches diameter of cylinder by 30 inches stroke, making 120 revolutions per minut% with reciprocating parts weighing 1200 pounds, the force required on the centres to impart and arrest the motion of these parts is 36"8 pounds per square inch of piston. In the smaller engines the force required for this purpose becomes, at the same speed of piston, considerably greater even than this. In the siz~ 6 inches diameter of cylinder by 12 inches stroke, with reciprocating parts weighing 100 pounds, it amounts on each centre to 54: pounds for each square inch of piston. Thus in these engines this reciprocating fly-wheel, at the speed of 600 feet per minute, absorbs, at the commencement of each stroke, from 3i" to 54: pounds of the force of the steam on each square inch of the piston, permitting it to become gradually eitbctive on the crank, and gives out this force again when that of the steam VOL. [,X.--TIIII~D SI~RIEs.--No, 5.---N,~V~:MI~I.:I~.,1870.
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317
balanced, the table may be readily moved by one man. tie is provided with an iron bar, one end of which fits to the bolt previously spoken of. He walks along the curb, pushing by means of this bar, and when he arrives at the desired track, immediately drives the bolt home, securing the table where wanted.
BELTING FACTSAND FIGURES, BY J. H. COOPER. (Continued from page 254.)
MR. F. W. B.~coN has kindly furnished me with the following: A certain 13-inch pulley un a shaft running 203 revolutions per minute carries a belt 2"25 inches wide, and drives a 20-inch pulley on a shaft 20 feet vertically above. The pulleys are smooth turned iron and the belt of single leather, with grain side to pulleys. This belt had been running a year or more, under a tension which was limited only by the strength of the lacing. It was used to convey two horse.power to the upper shaft, but was considered by the lessee to be unequal to the task, even when tightly drawn, and its adhesion increased by free application of rosin. It was admitted by both parties that the belt was worked to its fullest capacity. In order to ascertain the exact amount of work done by the belt, the following experiment was made : On the driven shaft above was a smooth turned iron pulley, 6"25 feet in circumference and 4-inch face ; over this was thrown a 3-inch leather belt, with grain side to pulley, and to its ends were attached unequal weights, such that t h e 2"25-inch belt was subjected to its maximum working power. These weights were 203 pounds and 2"25 pounds, Speed of friction pulley was taken at 132 revolutions per minute. Then we have 132 × 6"25 ~ 825 =_ velocity of 3-inch belt in feet per minute, and 825 X 200"75 5"018 ~ horse power of 3-inch ---33,-00b-- = belt. This is equivalent to a driving power of 41"1 square feet of belt per minute per horse.power. Rule for Horse-power of a Belt, from " Overman's Mechanics," p. 414. --One example will explain the table. It is assumed that belts
318
Civil and .~Tecltanica~ Engineering.
should have a speed of 25 to 30 feet per second, and that a 12-inch belt on a 3-feet pulley will transmit 10 horse-power. At 25 feet per second this means a surface velocity of 150 square feet per minute per horse power, which is a liberal allowance. The ~uarter turn belt applied to driving mill stones or upright sl~afts.--A is the driving pulley on horizontal shaft. B the driven pulley on mill spindle or upright shaft, c the tightener, which is placed at the proper angle Ibr receiving the belt from B and delivering it to A ; it has a short shaft running in bearings secured to a
frame which slides vertical!y in fixed grooves, and may be raised to tighten the belt for driving, or lowered to slacken the belt for stopping B, at pleasure. B is made wide and straight on the face to admit of motion in raising and lowering the stones, as well as to allow of lead of belt by the different positions of c, which are due to length and tightness of belt. A and o should be rounding on their faces. The cut shows the proper positions of the pulleys and shafts, and also gives good
working proportions, the particular's having been obtained from machinery in use. W e make a few extracts from a paper " O n the centrifugal force of bands in machinery," b y W. J. M. Rankine, in the Engineer for March 5, 1869, p. 165 : " I t is well known, through practical experience, that a belt for communicating motion between two pulleys requires a greater ten-
BeSting Facts and i'~gures.
3 !9
sion to prevent it from slipping when it runs at a high, than at a low speed. "Various suppositions have been made to account for this, such as that of the adhesion to the belt of a layer of air, which at a very high speed has not time to escape from between the belt and the pulley. But the real cause is simply the centrifugal force of the belt, which acts against its tension, and therefore slackens its grip of the pulleys. " ~'~ It can be proved from the elementary laws of dynamics, that if an endless band, of any figure whatsoever, runs at a given speed, the centrifugal force produces an uniform tension at each cross section of the band, equal to the weight of a piece of the band, whose length is twice the height from which a heavy body must fall, in order to acquire the velocity of the band. " In symbols, let w be the weight of a unit of length of the band ; v the speed at which it runs, and g the velocity produced by gravity in a second ( ~ 32"2 feet) ; then the centrifugal tension (as it may be ~UV 2
called) has the following value :-
g. " T h e effect on the band when in motion is, that at any given point, the tension which produces pressure and friction on the pulleys, or available tension (as it is called), is less than the total tension b y an amount equal to the centrifugal tension ; for this amount is employed in compelling the particles of the band to circulate in a closed or endless path. It is, of course, to the total tension that the strength of the band is to be adapted, therefore the transverse dimensions of a band for transmitting a given force must be greater for a high than for a low speed. "One of the most convenient ways of expressing the size of a band is by stating its weight per unit of length; for example, in pounds per running foot or in kilogrammes per metre. When the size is expressed thus, the corresponding way of expressing the intensity of any stress on the band is in lineal units of itself, such as feet or metres. Let b denote the greatest safe working tension on a band of a given kind, in units of its own length; so that w l is the amount of the safe working tension in units of weight. Let T be the amount of the available tension required at the driving side of the band for the transmission of' power, being usually from two to two and a half times the force to be transmitted. Then the total tension is w v~ T+ =wl. J
320
Civil and Mechanical Engineering.
Whence it is obvious that the required weight per unit of length is given by the following formu]a: T qAT--
V2.
g " F o r example, suppose that the band is a wire rope, that the greatest working tension is to be equivalent to the weight of 2900 feet of the rope, and that it is to run at 100 feet per second: then we have 1 ~ 2900 feet ; V2
--- ~ 310 feet ; g And consequently the weight per running foot of the rope required is : - T ~: ----- 2900--310 ~ 2590; Or about one-eighth part heavier than the rope required, for a speed so moderate as to make the centrifugal tension unimportant. " I n fixing the value of the greatest working tension on a wire rope, a proper deduction must of course be made for the stress produced by the bending of the wires round the pulleys. That stress is given in equivalent length of rope by the expression Ld iij-' where D is the diameter of the smallest pulley round which the rope passes, d the diameter of the wire of which the rope is made, and T. the modulus of elasticity of the wire, in length of itself, viz : about 8,000,000 feet, or 2,400,000 metres. That is to say, let 1 be length of the rope equivalent to the greatest safe working tension on a straight rope; 1t as before, the length equivalent to the actual greatest working tension, then l~d
1_~- ZL---D
" In the case of leathern belts, b may be estimated at about 660 feet, or 200 metres. 't In the case of a leather belt running at the rate of 100 feet per second, the weight per unit of length required, in order to exert a 660 66O given available tension, is increased in the ratio of 660--310 350 or to nearly double, as compared with that of a belt whose centrifugal force is unimportant. " T h e section,~l area of a leathern belt may be calculated approximately in square inches by multiplying the weight per running
A lien En2;rv?.
321
foot by 2"3 ; or in square millimetres, by multiplying tile weight in kilogrammes to the running metre b)~ 1000. " The ordinary thickness of a single belt being about 0"16 inch, or 4 millimetres, tile breadth may be deduced from the sectional area by dividing by that thickness. " The length (L) equivalent to the modulus of elasticity of a leathern belt, as calculated from Bevan's experiments, is about 23,000 feet, or 7000 metres. (To be Continued)
THE ALLEN ENGINE, By CI~X~LES T. PORTI~I~. (Continued from page 252.) IX an engine of 16 inches diameter of cylinder by 42 inches stroke, making 60 revolutions per minute~ and in which the reciprocating parts weigh 600 pounds, the force required to impart and arrest their motion is, at the centres, only 6 pounds on the square inch of piston, diminishing to nothing at the mid-stroke. I f the speed of piston ks increased from 420 feet, as above, to 600 feet per minute, by employing a stroke of 5 feet, and the weight of the reciprocating parts is increased to 1200 pounds, even then the force so absorbed and given out at the centres by these parts is only 18"~ pounds on the square inch of piston. In an Allen Engine, however, of 16 inches diameter of cylinder by 30 inches stroke, making 120 revolutions per minut% with reciprocating parts weighing 1200 pounds, the force required on the centres to impart and arrest the motion of these parts is 36"8 pounds per square inch of piston. In the smaller engines the force required for this purpose becomes, at the same speed of piston, considerably greater even than this. In the siz~ 6 inches diameter of cylinder by 12 inches stroke, with reciprocating parts weighing 100 pounds, it amounts on each centre to 54: pounds for each square inch of piston. Thus in these engines this reciprocating fly-wheel, at the speed of 600 feet per minute, absorbs, at the commencement of each stroke, from 3i" to 54: pounds of the force of the steam on each square inch of the piston, permitting it to become gradually eitbctive on the crank, and gives out this force again when that of the steam VOL. [,X.--TIIII~D SI~RIEs.--No, 5.---N,~V~:MI~I.:I~.,1870.
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