Modern Machine-Shop Practice Part 42

+-------------+-----------+-----------------------------+ | Average | Adjacent | DISTANCE BETWEEN THE CENTRES| | difference | steps, | OF CONE PULLEYS. | | between | whose +----+----+----+----+----+----+ | the | diffe- | | | | | | | | adjacent | rence is | 10 | 20 | 30 | 40 | 50 | 60 | | steps. | given in | i n c h e s. | | | table. | | | | | | | +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|0.90|0.95|0.97|0.98|0.98|0.98| | |2nd " 3rd|0.97|0.98|0.99|0.99|0.99|0.99| | 1 inch |3rd " 4th|1.03|1.02|1.01|1.01|1.01|1.01| | |4th " 5th|1.10|1.05|1.03|1.02|1.02|1.02| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|1.28|1.39|1.43|1.45|1.46|1.46| | |2nd " 3rd|1.43|1.46|1.48|1.48|1.48|1.49| | 1-1/2 inch |3rd " 4th|1.57|1.54|1.52|1.52|1.52|1.51| | |4th " 5th|1.72|1.61|1.57|1.55|1.54|1.54| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|1.61|1.81|1.87|1.90|1.92|1.93| | |2nd " 3rd|1.87|1.94|1.96|1.97|1.97|1.98| | 2 inches |3rd " 4th|2.13|2.06|2.04|2.03|2.03|2.02| | |4th " 5th|2.39|2.19|2.13|2.10|2.08|2.07| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|1.89|2.20|2.30|2.35|1.38|2.40| | |2nd " 3rd|2.30|2.40|2.43|2.45|2.46|2.47| | 2-1/2 inches|3rd " 4th|2.70|2.60|2.57|2.55|2.54|2.53| | |4th " 5th|3.11|2.80|2.70|2.65|2.62|2.60| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|2.10|2.57|2.71|2.78|2.83|2.86| | |2nd " 3rd|2.71|2.86|2.90|2.93|2.94|2.95| | 3 inches |3rd " 4th|3.29|3.14|3.10|3.07|3.06|3.05| | |4th " 5th|3.90|3.43|3.29|3.22|3.17|3.14| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd| |3.22|3.49|3.62|3.69|3.75| | |2nd " 3rd|3.48|3.74|3.83|3.87|3.90|3.91| | 4 inches |3rd " 4th|4.52|4.26|4.17|4.13|4.10|4.09| | |4th " 5th| |4.78|4.51|4.38|4.31|4.25| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd| |3.77|4.20|4.40|4.52|4.60| | |2nd " 3rd|4.19|4.60|4.73|4.80|4.84|4.87| | 5 inches |3rd " 4th|5.81|5.40|5.27|5.20|5.16|5.13| | |4th " 5th| |6.23|5.80|5.60|5.48|5.40| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd| |4.21|4.83|5.13|5.31|5.42| | |2nd " 3rd|4.82|5.42|5.62|5.71|5.77|5.81| | 6 inches |3rd " 4th|7.18|6.58|6.38|6.29|6.23|6.19| | |4th " 5th| |7.79|7.17|6.87|6.69|6.58| +-------------+-----------+----+----+----+----+----+----+

+-------------+-----------+-----------------------------+ | Average | Adjacent | DISTANCE BETWEEN THE CENTRES| | difference | steps, | OF CONE PULLEYS. | | between | whose +----+----+----+----+----+----+ | the | diffe- | | | | | | | | adjacent | rence is | 70 | 80 | 90 | 100| 120| 240| | steps. | given in | i n c h e s. | | | table. | | | | | | | +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|0.99|0.99|0.99|0.99|0.99|1.00| | |2nd " 3rd|0.99|1.00|1.00|1.00|1.00|1.00| | 1 inch |3rd " 4th|1.01|1.00|1.00|1.00|1.00|1.00| | |4th " 5th|1.01|1.01|1.01|1.01|1.01|1.00| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|1.47|1.47|1.48|1.48|1.48|1.49| | |2nd " 3rd|1.49|1.49|1.49|1.49|1.49|1.49| | 1-1/2 inch |3rd " 4th|1.51|1.51|1.51|1.51|1.51|1.51| | |4th " 5th|1.53|1.53|1.52|1.52|1.52|1.51| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|1.94|1.95|1.96|1.96|1.97|1.98| | |2nd " 3rd|1.98|1.98|1.99|1.99|1.99|1.99| | 2 inches |3rd " 4th|2.02|2.02|2.01|2.01|2.01|2.01| | |4th " 5th|2.06|2.05|2.04|2.04|2.03|2.02| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|2.41|2.42|2.43|2.44|2.45|2.47| | |2nd " 3rd|2.47|2.47|2.48|2.48|2.48|2.49| | 2-1/2 inches|3rd " 4th|2.53|2.53|2.52|2.52|2.52|2.51| | |4th " 5th|2.59|2.58|2.57|2.56|2.55|2.53| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|2.87|2.89|2.90|2.91|2.93|2.96| | |2nd " 3rd|2.96|2.96|2.97|2.97|2.98|2.99| | 3 inches |3rd " 4th|3.04|3.04|3.03|3.03|3.02|3.01| | |4th " 5th|3.13|3.11|3.10|3.09|3.07|3.04| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|3.78|3.81|3.83|3.84|3.87|3.94| | |2nd " 3rd|3.92|3.94|3.94|3.95|3.96|3.98| | 4 inches |3rd " 4th|4.08|4.06|4.06|4.05|4.04|4.02| | |4th " 5th|4.22|4.19|4.17|4.16|4.13|4.06| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|4.66|4.71|4.73|4.76|4.80|4.90| | |2nd " 3rd|4.89|4.90|4.91|4.92|4.93|4.96| | 5 inches |3rd " 4th|5.11|5.10|5.09|5.08|5.07|5.04| | |4th " 5th|5.34|5.29|5.27|5.24|5.20|5.10| +-------------+-----------+----+----+----+----+----+----+ | |1st and 2nd|5.51|5.57|5.62|5.66|5.71|5.86| | |2nd " 3rd|5.83|5.86|5.87|5.88|5.90|5.95| | 6 inches |3rd " 4th|6.17|6.14|6.13|6.12|6.10|6.05| | |4th " 5th|6.49|6.43|6.38|6.34|6.29|6.14| +-------------+-----------+----+----+----+----+----+----+

Before describing and applying the table, we will call attention to the term "effective" diameter. The effective radius--as is well known--extends from the centre of the pulley to the centre of the belt; the effective diameter, being twice this effective radius, must also equal the actual diameter plus thickness of belt.

The table is so arranged that the diameter (divided by distance between centres) of one step of a belted pair will always be found in the extreme right-hand column; while its companion step will be found on the same horizontal line, and in that vertical column of the table corresponding to the length of belt employed. For example, if column 14 of the table corresponded to the length of belt employed, some of the possible pairs of diameters would be as follows:

.7118 .5813 .42 .2164 .0474 .06 .24 .42 .60 .72

The upper row of this series of pairs being taken from column 14, and the lower row from the extreme right-hand column, the numbers in each pair being on the same horizontal line. If the distance between the centers of the pulleys were 60 ins. the effective diameters of the steps corresponding to the above pairs would be:

42.71 34.88 25.2 12.98 2.84 ins.

3.6 14.4 25.2 36.0 43.20

being obtained by multiplying the first series of pairs by 60; the length of belt which would be equally tight on each of these pairs would be 3.3195 60 ins. = 199.17 ins.

III.--TABLE FOR FINDING THE EFFECTIVE DIAMETERS OF THE STEPS OF CONE PULLEYS, WHEN THE PULLEYS ARE CONNECTED BY AN OPEN BELT AND ARE UNLIKE.

Each vertical cone of the table corresponds to a given length of belt, and the numbers in these columns are the required effective diameters of the steps when the centres of the pulleys are a Unit's distance apart.

+--------------------------------------------------------------+------+ | LENGTH OF BELT WHEN THE CENTRES OF THE CONE PULLEYS ARE A | | | UNIT'S DISTANCE APART. | | +------+------+------+------+------+------+------+------+------+ | |2.0942|2.1885|2.2827|2.3770|2.4712|2.5655|2.6597|2.7540|2.8482| | +------+------+------+------+------+------+------+------+------+ [A] | | =1= | =2= | =3= | =4= | =5= | =6= | =7= | =8= | =9= | | +------+------+------+------+------+------+------+------+------+------+ | .0594| .1177| .1750| .2313| .2867| .3413| .3950| .4479| .5000| 0.00 | | .03 | .0894| .1477| .2050| .2613| .3167| .3713| .4250| .4779| 0.03 | | | .06 | .1194| .1777| .2350| .2913| .3467| .4013| .4550| 0.06 | | | .0294| .09 | .1494| .2077| .2650| .3213| .3767| .4313| 0.09 | | | | .0594| .12 | .1794| .2377| .2950| .3513| .4067| 0.12 | | | | .0275| .0894| .15 | .2094| .2677| .3250| .3813| 0.15 | | | | | .0575| .1194| .18 | .2394| .2977| .3550| 0.18 | | | | | .0244| .0875| .1494| .21 | .2694| .3277| 0.21 | | | | | | .0544| .1175| .1794| .24 | .2994| 0.24 | | | | | | .0200| .0844| .1475| .2094| .27 | 0.27 | | | | | | | .0500| .1144| .1775| .2394| 0.30 | | | | | | | .0140| .0800| .1444| .2075| 0.33 | | | | | | | | .0440| .1100| .1744| 0.36 | | | | | | | | .0064| .0740| .1400| 0.39 | | | | | | | | | .0364| .1040| 0.42 | | | | | | | | | | .0664| 0.45 | | | | | | | | | | .0271| 0.48 | +------+------+------+------+------+------+------+------+------+------+

+--------------------------------------------------------------+------+ | LENGTH OF BELT WHEN THE CENTRES OF THE CONE PULLEYS ARE A | | | UNIT'S DISTANCE APART. | | +------+------+------+------+------+------+------+------+------+ | |2.9425|3.0367|3.1310|3.2252|3.3195|3.4137|3.5080|3.6022|3.6965| | |------+------+------+------+------+------+------+------+------+ [A] | | =10= | =11= | =12= | =13= | =14= | =15= | =16= | =17= | =18= | | +------+------+------+------+------+------+------+------+------+------+ | .5514| .6020| .6518| .7010| .7495| .7974| .8447| .8913| .9373| 0.00 | | .5300| .5814| .6320| .6818| .7310| .7795| .8274| .8747| .9213| 0.03 | | .5079| .5600| .6114| .6620| .7118| .7610| .8095| .8574| .9047| 0.06 | | .4850| .5379| .5900| .6414| .6920| .7418| .7910| .8395| .8874| 0.09 | | .4613| .5150| .5679| .6200| .6714| .7220| .7718| .8210| .8695| 0.12 | | .4367| .4913| .5450| .5979| .6500| .7014| .7520| .8018| .8510| 0.15 | | .4113| .4667| .5213| .5750| .6279| .6800| .7314| .7820| .8318| 0.18 | | .3850| .4413| .4967| .5513| .6050| .6579| .7100| .7614| .8120| 0.21 | | .3577| .4150| .4713| .5267| .5813| .6350| .6879| .7400| .7914| 0.24 | | .3294| .3877| .4450| .5013| .5567| .6113| .6650| .7179| .7700| 0.27 | | .30 | .3594| .4177| .4750| .5313| .5867| .6413| .6950| .7479| 0.30 | | .2694| .33 | .3894| .4477| .5050| .5613| .6167| .6713| .7250| 0.33 | | .2375| .2994| .36 | .4194| .4777| .5350| .5913| .6467| .7013| 0.36 | | .2044| .2675| .3294| .39 | .4494| .5077| .5650| .6213| .6767| 0.39 | | .1700| .2344| .2975| .3594| .42 | .4794| .5377| .5950| .6513| 0.42 | | .1340| .2000| .2644| .3275| .3894| .45 | .5094| .5677| .6250| 0.45 | | .0964| .1640| .2300| .2944| .3575| .4194| .48 | .5394| .5977| 0.48 | | .0571| .1264| .1940| .2600| .3244| .3875| .4494| .51 | .5694| 0.51 | | .0160| .0871| .1564| .2240| .2900| .3544| .4175| .4794| .54 | 0.54 | | | .0460| .1171| .1864| .2540| .3200| .3844| .4475| .5094| 0.57 | | | .0029| .0760| .1471| .2164| .2840| .3500| .4144| .4775| 0.60 | | | | .0329| .1060| .1771| .2464| .3140| .3800| .4444| 0.63 | | | | | .0629| .1360| .2071| .2764| .3440| .4100| 0.66 | | | | | .0174| .0929| .1660| .2371| .3064| .3740| 0.69 | | | | | | .0474| .1229| .1960| .2671| .3364| 0.72 | | | | | | | .0774| .1529| .2260| .2971| 0.75 | | | | | | | .0292| .1074| .1829| .2560| 0.78 | | | | | | | | .0592| .1374| .2129| 0.81 | | | | | | | | .0081| .0892| .1674| 0.84 | | | | | | | | | .0381| .1192| 0.87 | | | | | | | | | | .0681| 0.90 | | | | | | | | | | .0138| 0.93 | +------+------+------+------+------+------+------+------+------+------+

+--------------------------------------------------------------+------+ | LENGTH OF BELT WHEN THE CENTRES OF THE CONE PULLEYS ARE A | | | UNIT'S DISTANCE APART. | | +------+------+------+------+------+------+------+------+------+ | |3.7907|3.8850|3.9792|4.0735|4.1677|4.2620|4.3562|4.4504|4.5447| | |------+------+------+------+------+------+------+------+------+ [A] | | =19= | =20= | =21= | =22= | =23= | =24= | =25= | =26= | =27= | | +------+------+------+------+------+------+------+------+------+------+ | .9828|1.0277|1.0721|1.1159|1.1593|1.2021|1.2444|1.2861|1.3274| 0.00 | | .9673|1.0128|1.0577|1.1021|1.1459|1.1893|1.2321|1.2744|1.3161| 0.03 | | .9513| .9973|1.0428|1.0877|1.1321|1.1759|1.2193|1.2621|1.3044| 0.06 | | .9347| .9813|1.0273|1.0728|1.1177|1.1621|1.2059|1.2493|1.2921| 0.09 | | .9174| .9647|1.0113|1.0573|1.1028|1.1477|1.1921|1.2359|1.2793| 0.12 | | .8995| .9474| .9947|1.0413|1.0873|1.1328|1.1777|1.2221|1.2659| 0.15 | | .8810| .9295| .9774|1.0247|1.0713|1.1173|1.1628|1.2077|1.2521| 0.18 | | .8618| .9110| .9595|1.0074|1.0547|1.1013|1.1473|1.1928|1.2377| 0.21 | | .8420| .8918| .9410| .9895|1.0374|1.0847|1.1313|1.1773|1.2228| 0.24 | | .8214| .8720| .9218| .9710|1.0195|1.0674|1.1147|1.1613|1.2073| 0.27 | | .8000| .8514| .9020| .9518|1.0010|1.0495|1.0974|1.1447|1.1913| 0.30 | | .7779| .8300| .8814| .9320| .9818|1.0310|1.0795|1.1274|1.1747| 0.33 | | .7550| .8079| .8600| .9114| .9620|1.0118|1.0610|1.1095|1.1574| 0.36 | | .7313| .7850| .8379| .8900| .9414| .9920|1.0418|1.0910|1.1395| 0.39 | | .7067| .7613| .8150| .8679| .9200| .9714|1.0220|1.0718|1.1210| 0.42 | | .6813| .7367| .7913| .8450| .8979| .9500|1.0014|1.0520|1.1018| 0.45 | | .6550| .7113| .7667| .8213| .8750| .9279| .9800|1.0314|1.0820| 0.48 | | .6277| .6850| .7413| .7967| .8513| .9050| .9579|1.0100|1.0614| 0.51 | | .5994| .6577| .7150| .7713| .8267| .8813| .9350| .9879|1.0400| 0.54 | | .57 | .6294| .6877| .7450| .8013| .8567| .9113| .9650|1.0179| 0.57 | | .5394| .60 | .6594| .7177| .7750| .8313| .8867| .9413| .9950| 0.60 | | .5075| .5694| .63 | .6894| .7477| .8050| .8613| .9167| .9713| 0.63 | | .4744| .5375| .5994| .66 | .7194| .7777| .8350| .8913| .9467| 0.66 | | .4400| .5044| .5675| .6294| .69 | .7494| .8077| .8650| .9213| 0.69 | | .4040| .4700| .5344| .5975| .6594| .72 | .7794| .8377| .8950| 0.72 | | .3664| .4340| .5000| .5644| .6275| .6894| .75 | .8094| .8677| 0.75 | | .3271| .3964| .4640| .5300| .5944| .6575| .7194| .78 | .8394| 0.78 | | .2860| .3571| .4264| .4940| .5600| .6244| .6875| .7494| .81 | 0.81 | | .2429| .3160| .3871| .4564| .5240| .5900| .6544| .7175| .7794| 0.84 | | .1974| .2729| .3460| .4171| .4864| .5540| .6200| .6844| .7475| 0.87 | | .1492| .2274| .3029| .3760| .4471| .5164| .5840| .6500| .7144| 0.90 | | .0981| .1792| .2574| .3329| .4060| .4771| .5464| .6140| .6800| 0.93 | | .0438| .1281| .2092| .2874| .3629| .4360| .5071| .5764| .6440| 0.96 | | | .0738| .1581| .2392| .3174| .3929| .4660| .5371| .6064| 0.99 | | | .0157| .1038| .1881| .2692| .3474| .4229| .4960| .5671| 1.02 | | | | .0457| .1338| .2181| .2992| .3774| .4529| .5260| 1.05 | | | | | .0757| .1638| .2481| .3292| .4074| .4829| 1.08 | | | | | .0131| .1057| .1938| .2781| .3592| .4374| 1.11 | | | | | | .0431| .1357| .2238| .3081| .3892| 1.14 | | | | | | | .0731| .1657| .2538| .3381| 1.17 | | | | | | | .0050| .1031| .1957| .2838| 1.20 | | | | | | | | .0350| .1331| .2257| 1.23 | | | | | | | | | .0650| .1631| 1.26 | | | | | | | | | | .0950| 1.29 | | | | | | | | | | .0200| 1.32 | +------+------+------+------+------+------+------+------+------+------+

+-----------------------------------------+------+ | LENGTH OF BELT WHEN THE CENTRES OF THE | | |CONE PULLEYS ARE A UNIT'S DISTANCE APART.| | +------+------+------+------+------+------+ | |4.6389|4.7332|4.8274|4.9217|5.0159|5.1102| | |------+------+------+------+------+------+ [A] | | =28= | =29= | =30= | =31= | =32= | =33= | | +------+------+------+------+------+------+------+ |1.3682|1.4085|1.4484|1.4877|1.5266|1.5650| 0.00 | |1.3574|1.3982|1.4385|1.4784|1.5177|1.5566| 0.03 | |1.3461|1.3874|1.4282|1.4685|1.5084|1.5477| 0.06 | |1.3344|1.3761|1.4174|1.4582|1.4985|1.5384| 0.09 | |1.3221|1.3644|1.4061|1.4474|1.4882|1.5285| 0.12 | |1.3093|1.3521|1.3944|1.4361|1.4774|1.5182| 0.15 | |1.2959|1.3393|1.3821|1.4244|1.4661|1.5074| 0.18 | |1.2821|1.3259|1.3693|1.4121|1.4544|1.4961| 0.21 | |1.2677|1.3121|1.3559|1.3993|1.4421|1.4844| 0.24 | |1.2528|1.2977|1.3421|1.3859|1.4293|1.4721| 0.27 | |1.2373|1.2828|1.3277|1.3721|1.4159|1.4593| 0.30 | |1.2213|1.2673|1.3128|1.3577|1.4021|1.4459| 0.33 | |1.2047|1.2513|1.2973|1.3428|1.3877|1.4321| 0.36 | |1.1874|1.2347|1.2813|1.3273|1.3728|1.4177| 0.39 | |1.1695|1.2174|1.2647|1.3113|1.3573|1.4028| 0.42 | |1.1510|1.1995|1.2474|1.2947|1.3413|1.3873| 0.45 | |1.1318|1.1810|1.2295|1.2774|1.3247|1.3713| 0.48 | |1.1120|1.1618|1.2110|1.2595|1.3074|1.3547| 0.51 | |1.0914|1.1420|1.1918|1.2410|1.2895|1.3374| 0.54 | |1.0700|1.1214|1.1720|1.2218|1.2710|1.3195| 0.57 | |1.0479|1.1000|1.1514|1.2020|1.2518|1.3010| 0.60 | |1.0250|1.0779|1.1300|1.1814|1.2320|1.2818| 0.63 | |1.0013|1.0550|1.1079|1.1600|1.2114|1.2620| 0.66 | | .9767|1.0313|1.0850|1.1379|1.1900|1.2414| 0.69 | | .9513|1.0067|1.0613|1.1150|1.1679|1.2200| 0.72 | | .9250| .9813|1.0367|1.0913|1.1450|1.1979| 0.75 | | .8977| .9550|1.0113|1.0667|1.1213|1.1750| 0.78 | | .8694| .9277| .9850|1.0413|1.0967|1.1513| 0.81 | | .84 | .8994| .9577|1.0150|1.0713|1.1267| 0.84 | | .8094| .87 | .9294| .9877|1.0450|1.1013| 0.87 | | .7775| .8394| .90 | .9594|1.0177|1.0750| 0.90 | | .7444| .8075| .8694| .93 | .9894|1.0477| 0.93 | | .7100| .7744| .8375| .8994| .96 |1.0194| 0.96 | | .6740| .7400| .8044| .8675| .9294| .99 | 0.99 | | .6364| .7040| .7700| .8344| .8975| .9594| 1.02 | | .5971| .6664| .7340| .8000| .8644| .9275| 1.05 | | .5560| .6271| .6964| .7640| .8300| .8944| 1.08 | | .5129| .5860| .6571| .7264| .7940| .8600| 1.11 | | .4674| .5429| .6160| .6871| .7564| .8240| 1.14 | | .4192| .4974| .5729| .6460| .7171| .7864| 1.17 | | .3681| .4492| .5274| .6029| .6760| .7471| 1.20 | | .3138| .3981| .4792| .5574| .6329| .7060| 1.23 | | .2557| .3438| .4281| .5092| .5874| .6629| 1.26 | | .1931| .2857| .3738| .4581| .5392| .6174| 1.29 | | .1250| .2231| .3157| .4038| .4881| .5692| 1.32 | | .0500| .1550| .2531| .3457| .4338| .5181| 1.35 | | | .0800| .1850| .2831| .3757| .4638| 1.38 | | | | .1100| .2150| .3131| .4057| 1.41 | | | | .0255| .1400| .2450| .3431| 1.44 | | | | | .0555| .1700| .2750| 1.47 | | | | | | .0855| .2000| 1.50 | +------+------+------+------+------+------+------+

Legend: [A] = a.s.sumed diameter of steps, divided by distance between the centres of Cone Pulleys.

To get the actual diameters of these steps when thickness of belt = 7/32 = 0.22 in., we have simply to subtract 0.22 in. from the effective diameters just given, thus:

42.49 34.66 24.98 12.76 2.62 in.

3.38 14.18 24.98 35.78 42.98

would be the series of pairs of actual diameters.

In solving problems relating to the diameters of cone pulleys by means of the accompanying table, we must have, besides the distance between centres, sufficient data to determine the column representing the length of belt. The length of belt is seldom known because it is of small practical importance to know its exact length; but it may be estimated approximately, and then the determination of suitable diameters of the steps becomes an extremely simple matter, as may be seen from what has already preceded. When the length of the belt is not known, and has not been a.s.sumed, we indirectly prescribe the length of belt by a.s.suming the effective diameters of the two steps of a belted pair; thus, in the following Figure (561), the length of belt is prescribed when the distance A B, and any one of the pairs of steps D_{1}_d__{1}, D_{2}_d__{2}, D_{3}_d__{3} and D_{4}_d__{4} are given. We will show in the following examples how the length of belt and its corresponding column of diameter may be found when a pair of steps (like D_{1}_d__{1}), are given.

[Ill.u.s.tration: Fig. 561.]

EXAMPLE 1. Given the effective diameters

4.5 in. 9 in. 15 in. 21 in. on cone A, -- -- 15 in. -- " B,

and the distance between centres equal to 50 inches.

Required the remaining diameters on cone B.

Since in this example the steps of the given pair are equal, we look for 15/50 = 0.30, in the extreme right-hand column of table; we will find it in the 11th line from the top; now looking along this line for the diameter of the other step, = 15/50 = 0.30, we will find it in column 10; consequently the numbers of this column may be taken as the diameters of the steps which are the companions or partners of those in the extreme right-hand column.

We can now easily determine the remaining members of the pairs to which 4.5 in., 9 in., and 21 in. steps respectively belong. To find the partner of the 4.5 step, we find 4.5/50 = 0.09 in the right-hand column, and look along the horizontal line on which 0.09 is placed till we come to column 10, in which we will find the number 0.4850; 0.4850 50 in. = 24.25 in. will be the effective diameter of the companion to the 4.5 in.

step.

To find the partner to the 9 in. step, we proceed as before, looking for 9/50 = 0.18 in the right-hand column, and then along the horizontal line of 0.18 to column 10, then will 0.4113 50 in. = 20.57 in. be the required companion to the 9 in. step of cone A.

In like manner for the partner of the 21 in. step we get 0.1700 50 in.

= 8.5 in. The effective diameter therefore will be,

4.5 in. 9 in. 15 in. 21 in. on cone A, 24.25 20.57 15 in. 8.5 " B.

If the thickness of belt employed were 0.25 in. the _actual_ diameters of steps would be,

4.25 8.75 14.75 20.75 on cone A, 24.00 20.32 14.75 8.25 " B,

and the length of belt would be 2.9425 50 = 147.125 in.

EXAMPLE 2. Given the effective diameters

6 in. 12 in. 18 in. 24 in. on cone A, 30 in. -- -- -- " B,

and the distance between centres = 40 in.

Required the unknown diameters on cone B.

We must, as before, first find the vertical column corresponding to the length of belt which joins the pair of steps 6 in/30 in. We find the number 6/40 = .15 in the right-hand column, and then look along its horizontal line for its partner 30/40 = 0.75. Since we do not find any number exactly equal to .7500, we must interpolate. For the benefit of those not familiar with the method of interpolation we will give in detail the method of finding intermediate columns of the table. On the aforesaid horizontal line we find in column 16 a number 0.7520, larger than the required 0.7500, and in column 15 a number 0.7014, smaller than 0.7500; evidently the intermediate column, containing the required 0.7500, must lie between columns 16 and 15. To find how far the required column is from column 16, we subtract as follows:

0.7520 0.7520 0.7500 0.7014 ------ ------ .0020 0.0506

then the fraction .0020/.00506 = 0.04 nearly will represent the position of the required intermediate column; namely, that its distance from column 16 is about 4/100 of the distance between the adjacent columns, 15 and 16.

To find other numbers in this intermediate column we have only to multiply the difference between the adjacent numbers of columns 16 and 15 by 0.04, and subtract the product from the number in column 16. But it is not necessary to find as many numbers of the intermediate columns as are contained in either of the adjacent columns; it is only necessary to find as many numbers as there are steps in each of the cone pulleys.

We will now ill.u.s.trate what has preceded, by finding the partner to the 12 in. step of cone A. Find, as before, the horizontal line corresponding to 12/40 = 0.30, then take the difference between the numbers 0.6413 and 0.5867 of columns 16 and 15, and multiply this difference, 0.0546, by 0.04; this product = 0.0022 subtracted from 0.6413, will give 0.6391, a number of the intermediate columns corresponding to the length of belt of the present problem. Multiplying by the distance between the axes = 40 in. we get 0.6391 40 = 25.56, for the diameter of the step of cone B which is partner to the 12 in.

step of cone A.

To find the companion to the 18 in. step, we proceed in the same manner, looking for the horizontal line 18/40 = 0.45, and interpolating as follows:

0.5094 - (0.5094 - 0.4500) 0.04 = 0.5070.

Consequently, 0.5070 40 in. = 20.28 in. will be the required partner of the 18 in. step.

In like manner, for the 24 in. step, we have

0.3500 - (0.3500 - 0.2840) 0.04 = 0.3474, and 0.3474 40 = 13.90.

The effective diameters are therefore

6 in. 12 in. 18 in. 24 in. on cone A.

30 25.56 20.28 13.9 " B.