Volume 12 (1866) (428 pages)

The Mining and Scientific. Bress.9 259. has given Professor Rankine, Weisbach, Johnson, Sehicle, the authors of “ Des Ingenieur’s Taschenbuch,’ ‘ and MM. Armeugaud and Amouroux what the “ Robber Kitten” in the dreary wood gave the cock, “an awful shock 1 shack, shock, shock P Dat if he ins not accomplished what he proposed, be will.at least bnve the cansolition left to him Itt Knowing that “it is not the first time in the bistory of human atfairs ‘that the vigor and! success of the war have nut quite come up to the lofty’ and soundiog phrase of the manifesto.” i By npplying the same! scale as “adopted rbove, in reference to the relative grinding effects of weir formed plates to. the Saath “obtained, on pages 107," 108 : and 109 of the « Quartz Operator’s Daud Book,” apd we sball havo for the griuding effect of Flat plutes or pivots.. Hpac > Conical plates or pivots. . Pca ewes Tructory conoidal plates or pivots. .1,0000 These resulta differ from those obtained by the authors cited. The reason of this difference is, that the authors in deterinining the mpment of friction for differently formed pivots, have not taken into account the wear to the pivot itself, Thnt is, they hnve presumed friction to take place without, wear to the' pivots, depending upon” unguents to prevent ‘the pivot and step coming in contact. Professor Rankine on the subject of ungnents, says: “Three results in tbe preeeding table, Nos. 16, 17 and 18, have reference to smooth, firm surfaces of any kind, greased or lubricated to such an extent that the friction depends chiefly on the continual supply of unguent, and not sensibly on tbe nature of the solid surfaces; and this ought almost always tn be the case in maehincry.”» See also quétation ereinbefore given’ from the same gather ' ‘on tbe subject of . a Scheile’ 3 antifriction, curve.” SD we 8) ca miericare! ‘Tf the pivot wid step comic in contact so as to grind, Tcoticvive but two cases as to Hat and conical Pivots,’ compared to ivactory-formed in which the nuthors. cited’ are strictly correct; one in which all the surfaces are infinitely hard, and the other iu which the hardness of allat the circumference is the same, and deereases in the flat and conical pivots toward’ the center in the rutio,of the decrease of their, radii. See Hand’ Book: page 107. But if they do not come in contaet, heing as they should prevented hy proper unguent, then for’ most practical purposes the’ authors'cited are correet. : Comparing results it will be found . that the authorities, cited, and the Hand Book : argree as.to.the.ratio of the. moments of friction.or grinding. effects between the flat and conical pivots or plates, thus, > 129000 51,8590 5 7",6687,2 TABBY But Mr. Goody dar finds no such ratio. He tells us with the utmost assurance that the grinding effect of platie plates is one, that of conical, one, that of tractory conoidal, Qne; “that it is one, one, one, fall the way down,” -lilke the feks on which, otber distinguished philosopbers nverred the earth to stand. 1h 8) ee : Again, to prove the’ truthfulness of the Baietions in the “ Quartz Operators Wand Book,”, ” relative to the grinding effects of plane and iragtory “conoidal plates. Conceive a plate having a plane grinding surface, to be made up of an indefinite number Of independent concentrie rings, all of the same weight and thickness, and let their grinding effects be measured. in the plate on which they revolve. It is now evident, if, each ring be reyolyed independent of the others once about its axis or. center, that‘all will wear into the stationary plate at’ right angles to its face, to the same depth ; and hence the secondary surface to which tbey shall have arrived, will also be a plane. . Te _ ae : * But if there is no tendency of the rings to pass each other up or down, in other words, in a direction parallel with the axis of revolution, then a ‘solid of the same; dimensions and w. eight, would give.the same result. Let us next determine the form and contents of this solid. It ié evident that a ring of half the radius of anotber must be of twice the hight, in order to weigh as much. So a ring of, one-third, one-fourth, one-tenth, one-hundredth, or one-thousandth, etc. of the radius of another, must be three, four, ten,'one hundred, or one thousand, etc., times the bight in order: to weigh as much, The hight of these rings may be determined. relatively hy, diyiding the constant radius ojo the aos, of — ring . whose hight is sought. The thickness ofeach ringistn be supposed infinitely small. The contents, since the rings nre all of the same ~ weight, are rendily found by maltiplying the product n¢ the greater or constant radius, and its circumference by the hight of the ring (outer). Call this plate (A). Now the contents of aplane plate of uniform thickness, and of the same radius and hight a3 the outer ring in the solid above, are equal to one-half the product of the radius nnd circumference miultiplicd by the hight. Call this plate(B), We thus perceive that the Poel or weight nf plate, (3B) is but half the contents or weight of plate (A).And'this plate (B) being rigid,’as to its parts, would liave but half the grinding effect of plate '(A). If the grinding effect of plate (A) be a unit (1,0000), that of (B) will be (,5000). : Now the grinding effect of plnte (A) has been cn to be uniform like that ofthe tractory. That is, no portion of the plate impedes or increases the grinding effect of nny other portion of the plate. 1f equnlly thin rings of the same radius and hight be taken at the: cireumference of each plate, nnd revolved about their respective axes, their grinding effects will be equal. Ench, will wear into the stationary plata. to the same deptb. But ' that the grinding effect of the outer ring, taken in the tractory plate, shall neitherbe impeded nor increased by tbe balance of the plate, it follows tbat the vertical hight at nll points of the plate, parallel ub the axis, is equal to the hight of the outer ring. co i : The contents of a plate of this form are thé same “as the contents of a plane plate of equal radius and higbt,. See Hand Book, page 105. But tbe contents of the pline plate bave been shown ual to half the contents of plate (A): ge we ae il The* ‘contents of the ‘hollow’ tractory plate are then equal t to half the ‘contents of plate(A). But the. grind: ing effects of the tractory, have becn shown equal to the grinding effects of plate: C95 that is a unit (1. 0000). While the grinding effects of the plane plate, of equal radius and weight have been ‘shown equal io'one-half of this, to wit, (,5000). This is precisely the same ratio as, found in the “ Quartz Operator’s Hand Book.” ° * As to the assumption in the above discussion, that the grinding effect of any portion of-the tractory plate is not impeded nor increased by wie Ee portion: of the . plate, . I do notunderstand ‘that there ‘is any question. On this point,’ as already quoted, MM. Ar remengaud & Amouroux say: « ‘Considering the truncated cone!of the: .stop-cock plug, to be divided into a series of infinitely short lengths, -he (Schielo) ‘proposed to take a more obtuse cone for” each lar ger ‘portion, and in such progression that it would require equal pressure for every portion a of. the. surface to, causea uniform sinking of the plug in the course of wear ;” also,“ perfect uniformity of wear-in the -grinding surfaces is attained by the use of the ‘curved face.” Professor Ranking says: « "The wear of the surfaces i 1s ‘uniform at every point, so that they always fit each other , accurately, and the pressure is always uniformly distributed.” Weishach, in determining the, moment of friction of pivots or. toes” of uniform wear, says,:, “Tbe. moment has to be :the samé on all parts of the, foe, if we wish to have equal wear of the toe and its journal, consequently tbe tangent for the whole generating curve of tbe toe has to he the same, a” —that is, the curve is the tractory. The ‘demonstrations of the other authors hereinbefore cited, agree perfectly with the préceding quotations. And’ ‘T am happy for onee to find, if I understand Mr. Goodycar correctly, that he assumes the same thing. At least his demon-, stration is based on it. : But as to his attempted demonstrations of the grinding ‘effects of plane and conical plates, it is quite evident . that he was either ‘knowing too little, or too much. , On page 178 of the Minne anp Screntiric PRESS, he says: “ Let, now, the actual pressure per unit, of surface at the circumfereuice when the plates are in motion be represented by P; represent also the pressure at. any point whose radius is y, by P’” Then if a denote the radius of tbe plates, we have y , (62) ae paper _ i vy -On ey premises he finds < grinding ‘effect of, plane plates to he “27? Pa.” « And I will add that the oat is Bi fetiy deduced from. the premises. “ Taking Mr. Goodyear’s premises, let us see what would-be'the contents of the plate vine grinding effect . would be “ 2%? Pa®.” yet ot Pacts ey Py Putting V = the volume on solid contents, /and wer have dV = 2ii Pady. * a Taking the integral between 0 and a, and we have * Vela Now, the solid contents or rolume nf a plane picts whose radius is a, and whose uniform pressure or hight, is represented by P, ike " at } ees And this (” Pa) ~ the volume or size of the platewhich Mr. Goodyenr tells the public: bas for its grinding effect (“ 2? Pa”), when, in fact, as shown abnve, it is onlyhalf the volume or size of the plate which he has: employed in his calculatinns to obtain that result. Had he told the whole trath, lia would have given thé] same .result ns found in the Hand .Bonk,-page 108, which, by word and farmula aad protended: demnnstra+ tion, he has denounced a “ fallacy.” It is said that: “the art of all arts is to appear artless.” Extimating by: this standard, the pnim must be awarded to Mr. Goode: yenr for having produced the great, niasterpiece nf art.» The results which he obtains on page 194, Equation: 72, set forth to be the grinding effect nf conical plates,» and the manner of obtaining these results, are open to still severer criticisms. There, after committing an error similar to the one in relation to plane plntes, he “»with’ one fell swoop lops off” the conie form and substitutes the plane. Having now attained his plane plates nf double the weight proposed to be used at first, he all ati once becomes highly consistent, and has them necomplish the same grinding effect as his other plane plates of the same radius nnd weight. By reference to the Hand Book, page 110, and to ‘the formulas nf Weisbach and * Des Ingenieurs Taschenhuch,” hereinbefore quoted, it will he found that those authors were less artful ;’ that they could not strip the cone of its conic form ; nor, by any “ agamento, presto change,” double its real weight, and still-hate it retnin its weight and accomplish as a cone the same. grinding effect as a plane plate of equal size and weight. This will be found no exaggeration by any one who will carefully examine Mr. Goodyear’a: calculations and deductions relative to the grinding effects of conical plates. ‘If this be the way in which Mr. Goodyear, as the publie’s benefactor, shows up the “ fallacies” nf authors, it may well be doubted whether he will challenge very much of, the” respect: of mathematicians, mechanics and miners, saying nothing of the confidence of the public, 1) Let us now turn our nttention to the grinding . effects of plane, conical, and tractory conoidal rings, as ,prac-. tically employed in grinders andamalgamators. On this point Iam frank to admit that the “Hand Book is not. entirely correct, and Iam happy of ‘the opportunity of making the corrections myself. I can but regard myself saved in this respect, as was Rome at the time when the sentinels -hnd fallen asleep. . But before making the: corrections, let us compare the results obtained by Mr. Goodyear, to those of the nuthors hitherto quoted.) Mr. Goodyear, in the Minrnc anp Screntiric Press, page 210, gives for the-grinding, effects of rings, whose inner radii ate respectively one-third of the otter radii,’ as follows : -1 1 (78).” Plane Tings 2.4.6.6. “(80)” Conical rings.. ee ene 2) -<llah ' (81) 2 Tractory (conoidal.'>.. io All the same—the “ irreptvssible” ‘unit ratio still prevailing. It will be observed that’ he finds the grinding effect of the ring in each case to be less than that of the entire plate from which it was taken. «This is only true in the case of the tractory conoidal plate. The grinding effect of the respective plates entire corresponding to theabove numbers, is *.(2),” as will be seen by referring to his formulas (64,”. (72)” and “ (76).” Turning to “ Weisbach’s. Mechanics,” page 313, and to Professor Rankine’s “Applied ect niana page ‘617, and we find the following formula in each for ae moment! of friction of the flat ring re or plate, viz. . : prs pers bs oat td @ =) a In which 1, f, R andr have the same values as. in es mulas (1), (2) and (3). oa LL OE ITG IE , t é Oey . ' ~§ » Putting: r= re oa bied we have > ys ’ : Ts 78 : a : MN = __— : He a 108 ’ ond which, expressed decimally, is 7222. ‘This we perceive to be greater «sa the moment of friction for the whole plate or pivot, which, as shown, is’ 6667. Taking now the: formula for the moment of friction of the sie ring pivot, viz. : a) and anlage as bes , cB ee ral sud ily cal: =H 2LfR = “5 sina faye fore, the hice of tbe cone equal to one-half the cnn~» stant radius), and we have fnr the moment ofl frictionthe conical ring ,8074. a oe ‘All agree (including Mr. Goodyear). dia the monet of friction of the tractoroidal ring of equal weight and radii to the above is ,8889. » Taking “the snme unit a3 adopted by ‘Mr. Gnodyear,

The Mining and Scientific. Bress.9 259. has given Professor Rankine, Weisbach, Johnson, Sehicle, the authors of “ Des Ingenieur’s Taschenbuch,’ ‘ and MM. Armeugaud and Amouroux what the “ Robber Kitten” in the dreary wood gave the cock, “an awful shock 1 shack, shock, shock P Dat if he ins not accomplished what he proposed, be will.at least bnve the cansolition left to him Itt Knowing that “it is not the first time in the bistory of human atfairs ‘that the vigor and! success of the war have nut quite come up to the lofty’ and soundiog phrase of the manifesto.” i By npplying the same! scale as “adopted rbove, in reference to the relative grinding effects of weir formed plates to. the Saath “obtained, on pages 107," 108 : and 109 of the « Quartz Operator’s Daud Book,” apd we sball havo for the griuding effect of Flat plutes or pivots.... Hpac > Conical plates or pivots.. . Pca ewes Tructory conoidal plates or pivots.. .1,0000 These resulta differ from those obtained by the authors cited. The reason of this difference is, that the authors in deterinining the mpment of friction for differently formed pivots, have not taken into account the wear to the pivot itself, Thnt is, they hnve presumed friction to take place without, wear to the' pivots, depending upon” unguents to prevent ‘the pivot and step coming in contact. Professor Rankine on the subject of ungnents, says: “Three results in tbe preeeding table, Nos. 16, 17 and 18, have reference to smooth, firm surfaces of any kind, greased or lubricated to such an extent that the friction depends chiefly on the continual supply of unguent, and not sensibly on tbe nature of the solid surfaces; and this ought almost always tn be the case in maehincry.”» See also quétation ereinbefore given’ from the same gather ' ‘on tbe subject of . a Scheile’ 3 antifriction, curve.” SD we 8) ca miericare! ‘Tf the pivot wid step comic in contact so as to grind, Tcoticvive but two cases as to Hat and conical Pivots,’ compared to ivactory-formed in which the nuthors. cited’ are strictly correct; one in which all the surfaces are infinitely hard, and the other iu which the hardness of allat the circumference is the same, and deereases in the flat and conical pivots toward’ the center in the rutio,of the decrease of their, radii. See Hand’ Book: page 107. But if they do not come in contaet, heing as they should prevented hy proper unguent, then for’ most practical purposes the’ authors'cited are correet. : Comparing results it will be found . that the authorities, cited, and the Hand Book : argree as.to.the.ratio of the. moments of friction.or grinding. effects between the flat and conical pivots or plates, thus, > 129000 51,8590 5 7",6687,2 TABBY But Mr. Goody dar finds no such ratio. He tells us with the utmost assurance that the grinding effect of platie plates is one, that of conical, one, that of tractory conoidal, Qne; “that it is one, one, one, fall the way down,” -lilke the feks on which, otber distinguished philosopbers nverred the earth to stand. 1h 8) ee : Again, to prove the’ truthfulness of the Baietions in the “ Quartz Operators Wand Book,”, ” relative to the grinding effects of plane and iragtory “conoidal plates. Conceive a plate having a plane grinding surface, to be made up of an indefinite number Of independent concentrie rings, all of the same weight and thickness, and let their grinding effects be measured. in the plate on which they revolve. It is now evident, if, each ring be reyolyed independent of the others once about its axis or. center, that‘all will wear into the stationary plate at’ right angles to its face, to the same depth ; and hence the secondary surface to which tbey shall have arrived, will also be a plane. . Te _ ae : * But if there is no tendency of the rings to pass each other up or down, in other words, in a direction parallel with the axis of revolution, then a ‘solid of the same; dimensions and w. eight, would give.the same result. Let us next determine the form and contents of this solid. It ié evident that a ring of half the radius of anotber must be of twice the hight, in order to weigh as much. So a ring of, one-third, one-fourth, one-tenth, one-hundredth, or one-thousandth, etc. of the radius of another, must be three, four, ten,'one hundred, or one thousand, etc., times the bight in order: to weigh as much, The hight of these rings may be determined. relatively hy, diyiding the constant radius ojo the aos, of — ring . whose hight is sought. The thickness ofeach ringistn be supposed infinitely small. The contents, since the rings nre all of the same ~ weight, are rendily found by maltiplying the product n¢ the greater or constant radius, and its circumference by the hight of the ring (outer). Call this plate (A). Now the contents of aplane plate of uniform thickness, and of the same radius and hight a3 the outer ring in the solid above, are equal to one-half the product of the radius nnd circumference miultiplicd by the hight. Call this plate(B), We thus perceive that the Poel or weight nf plate, (3B) is but half the contents or weight of plate (A).And'this plate (B) being rigid,’as to its parts, would liave but half the grinding effect of plate '(A). If the grinding effect of plate (A) be a unit (1,0000), that of (B) will be (,5000). : Now the grinding effect of plnte (A) has been cn to be uniform like that ofthe tractory. That is, no portion of the plate impedes or increases the grinding effect of nny other portion of the plate. 1f equnlly thin rings of the same radius and hight be taken at the: cireumference of each plate, nnd revolved about their respective axes, their grinding effects will be equal. Ench, will wear into the stationary plata. to the same deptb. But ' that the grinding effect of the outer ring, taken in the tractory plate, shall neitherbe impeded nor increased by tbe balance of the plate, it follows tbat the vertical hight at nll points of the plate, parallel ub the axis, is equal to the hight of the outer ring. co i : The contents of a plate of this form are thé same “as the contents of a plane plate of equal radius and higbt,. See Hand Book, page 105. But tbe contents of the pline plate bave been shown ual to half the contents of plate (A): ge we ae il The* ‘contents of the ‘hollow’ tractory plate are then equal t to half the ‘contents of plate(A). But the. grind: ing effects of the tractory, have becn shown equal to the grinding effects of plate: C95 that is a unit (1. 0000). While the grinding effects of the plane plate, of equal radius and weight have been ‘shown equal io'one-half of this, to wit, (,5000). This is precisely the same ratio as, found in the “ Quartz Operator’s Hand Book.” ° * As to the assumption in the above discussion, that the grinding effect of any portion of-the tractory plate is not impeded nor increased by wie Ee portion: of the . plate, . I do notunderstand ‘that there ‘is any question. On this point,’ as already quoted, MM. Ar remengaud & Amouroux say: « ‘Considering the truncated cone!of the: .stop-cock plug, to be divided into a series of infinitely short lengths, -he (Schielo) ‘proposed to take a more obtuse cone for” each lar ger ‘portion, and in such progression that it would require equal pressure for every portion a of. the. surface to, causea uniform sinking of the plug in the course of wear ;” also,“ perfect uniformity of wear-in the -grinding surfaces is attained by the use of the ‘curved face.” Professor Ranking says: « "The wear of the surfaces i 1s ‘uniform at every point, so that they always fit each other , accurately, and the pressure

is always uniformly distributed.” Weishach, in determining the, moment of friction of pivots or. toes” of uniform wear, says,:, “Tbe. moment has to be :the samé on all parts of the, foe, if we wish to have equal wear of the toe and its journal, consequently tbe tangent for the whole generating curve of tbe toe has to he the same, a” —that is, the curve is the tractory. The ‘demonstrations of the other authors hereinbefore cited, agree perfectly with the préceding quotations. And’ ‘T am happy for onee to find, if I understand Mr. Goodycar correctly, that he assumes the same thing. At least his demon-, stration is based on it. : But as to his attempted demonstrations of the grinding ‘effects of plane and conical plates, it is quite evident . that he was either ‘knowing too little, or too much. , On page 178 of the Minne anp Screntiric PRESS, he says: “ Let, now, the actual pressure per unit, of surface at the circumfereuice when the plates are in motion be represented by P; represent also the pressure at. any point whose radius is y, by P’” Then if a denote the radius of tbe plates, we have y , (62) ae paper _ i vy -On ey premises he finds < grinding ‘effect of, plane plates to he “27? Pa.” « And I will add that the oat is Bi fetiy deduced from. the premises. “ Taking Mr. Goodyear’s premises, let us see what would-be'the contents of the plate vine grinding effect . would be “ 2%? Pa®.” yet ot Pacts ey Py Putting V = the volume on solid contents, /and wer have dV = 2ii Pady. * a Taking the integral between 0 and a, and we have * Vela Now, the solid contents or rolume nf a plane picts whose radius is a, and whose uniform pressure or hight, is represented by P, ike " at } ees And this (” Pa) ~ the volume or size of the platewhich Mr. Goodyenr tells the public: bas for its grinding effect (“ 2? Pa”), when, in fact, as shown abnve, it is onlyhalf the volume or size of the plate which he has: employed in his calculatinns to obtain that result. Had he told the whole trath, lia would have given thé] same .result ns found in the Hand .Bonk,-page 108, which, by word and farmula aad protended: demnnstra+ tion, he has denounced a “ fallacy.” It is said that: “the art of all arts is to appear artless.” Extimating by: this standard, the pnim must be awarded to Mr. Goode: yenr for having produced the great, niasterpiece nf art.» The results which he obtains on page 194, Equation: 72, set forth to be the grinding effect nf conical plates,» and the manner of obtaining these results, are open to still severer criticisms. There, after committing an error similar to the one in relation to plane plntes, he “»with’ one fell swoop lops off” the conie form and substitutes the plane. Having now attained his plane plates nf double the weight proposed to be used at first, he all ati once becomes highly consistent, and has them necomplish the same grinding effect as his other plane plates of the same radius nnd weight. By reference to the Hand Book, page 110, and to ‘the formulas nf Weisbach and * Des Ingenieurs Taschenhuch,” hereinbefore quoted, it will he found that those authors were less artful ;’ that they could not strip the cone of its conic form ; nor, by any “ agamento, presto change,” double its real weight, and still-hate it retnin its weight and accomplish as a cone the same. grinding effect as a plane plate of equal size and weight. This will be found no exaggeration by any one who will carefully examine Mr. Goodyear’a: calculations and deductions relative to the grinding effects of conical plates. ‘If this be the way in which Mr. Goodyear, as the publie’s benefactor, shows up the “ fallacies” nf authors, it may well be doubted whether he will challenge very much of, the” respect: of mathematicians, mechanics and miners, saying nothing of the confidence of the public, 1) Let us now turn our nttention to the grinding . effects of plane, conical, and tractory conoidal rings, as ,prac-. tically employed in grinders andamalgamators. On this point Iam frank to admit that the “Hand Book is not. entirely correct, and Iam happy of ‘the opportunity of making the corrections myself. I can but regard myself saved in this respect, as was Rome at the time when the sentinels -hnd fallen asleep. . But before making the: corrections, let us compare the results obtained by Mr. Goodyear, to those of the nuthors hitherto quoted.) Mr. Goodyear, in the Minrnc anp Screntiric Press, page 210, gives for the-grinding, effects of rings, whose inner radii ate respectively one-third of the otter radii,’ as follows : -1 1 (78).” Plane Tings 2.4.6.6. “(80)” Conical rings.... ee ene 2) -<llah ' (81) 2 Tractory (conoidal.'>... io All the same—the “ irreptvssible” ‘unit ratio still prevailing. It will be observed that’ he finds the grinding effect of the ring in each case to be less than that of the entire plate from which it was taken. «This is only true in the case of the tractory conoidal plate. The grinding effect of the respective plates entire corresponding to theabove numbers, is *.(2),” as will be seen by referring to his formulas (64,”. (72)” and “ (76).” Turning to “ Weisbach’s. Mechanics,” page 313, and to Professor Rankine’s “Applied ect niana page ‘617, and we find the following formula in each for ae moment! of friction of the flat ring re or plate, viz. . : prs pers bs oat td @ =) a In which 1, f, R andr have the same values as. in es mulas (1), (2) and (3). oa LL OE ITG IE , t é Oey . ' ~§ » Putting: r= re oa bied we have > ys ’ : Ts 78 : a : MN = __— : He a 108 ’ ond which, expressed decimally, is 7222. ‘This we perceive to be greater «sa the moment of friction for the whole plate or pivot, which, as shown, is’ 6667. Taking now the: formula for the moment of friction of the sie ring pivot, viz. : a) and anlage as bes , cB ee ral sud ily cal: =H 2LfR = “5 sina faye fore, the hice of tbe cone equal to one-half the cnn~» stant radius), and we have fnr the moment ofl frictionthe conical ring ,8074. a oe ‘All agree (including Mr. Goodyear). dia the monet of friction of the tractoroidal ring of equal weight and radii to the above is ,8889. » Taking “the snme unit a3 adopted by ‘Mr. Gnodyear,