Chap. XI.
MECHANICS.
943
for in reality the tangent of the angle must be to the radius as one diameter to the other.Hence we see that the bevel of the wheels must be such, if we suppose their shafts pro-duced, till they meet a line drawn from that point to the meeting of the circumference ofthe wheels will determine the angle or bevel of their edges, so that they will work freelyand equally together. When the acting faces of the teeth on the large wheel have the formof a spherical epicycloid, whose generating circle is the primitive circumference of thepinion, and whose base is the primitive circumference of the wheel, and if they drive thesmaller wheel by acting upon infinitely small pins in its circumference, the motion of bothwill be found uniform. If the acting faces of the teeth of the large wheel have the form ofspherical epicycloids, whose generating arch has a radius equal to half the primitive radiusof the smaller wheel, and whose base is the primitive circumference of the wheel, and ifthey drive the small wheel or pinion by acting upon rectilineal leaves directed to itscentre, the motion will be uniform. If a bevelled wheel drives a conical lantern withconical spindles, it will be done uniformly, when the acting faces of the teeth of the bevelledwheel have the form of a curve drawn parallel to the spherical epicycloid and distant fromit by the radius of the spindle.
Strength of the teeth of wheels may be calculated upon the principle of a beam fixed atone end, and sustaining pressure at the other; when the pressure at the pitch line is given,multiply it by the length in inches of the tooth to this line, and divide the product by2544 times the square of the thickness, and the quotient will be the breadth in inches;it is usual in practice to make the breadth of the wheel and teeth two or three times thepitch of the teeth.
The table gives the radii of wheels, the pitch being an inch, and the teeth of variousnumbers. When it is required to find the radius for any other pitch, it is only ne-cessary to multiply the number in the tables by the proposed pitch of the wheel, and theproduct will be the primitive or proportionate radius, viz. the radius measured from thecentre to the pitch line.
The table is founded upon the following principles: divide 360 degrees ljv twice thenumber of teeth, and it will give half the angle the pitch subtends at the centre, and thecosecant of this angle to a radius equal to half the pitch is the radius sought, thus: —
. . . . 180°
Primitive radius equal to half p x cosecant-=-^-.
r number of teeth
To determine the diameter of any wheel where the pitch and number of teeth are given,it is only necessary to multiply the number of the teeth by the pitch, and divide thisproduct by 3T416.
1 0
1
2
3
4
5
6
7
8
9
10
1 -668
1 -774
1-932
2-089
2-247
2-405
2*563
2-721
2*879
3-038
20
3-196
3 *355
3-513
3-672
3-830
3-989
4-148
4*307
4*465
4-624
30
4-783
4-942
5-101
5-260
5-419
5-578
5-737
5-896
6-055
6-214
40
6-373
6-532
6-643
6-850
7 009
7-168
7-327
7-486
7 '695
7-804
50
7-963
8-122
8-231
8-440
8-599
8-753
8-962
9*076
9*235
9*399
60
9-553
9-712
9-872
10 031
10-190
10-349
10-508
10-662
10-826
10-935
70
11-144
11 -303
11-463
11-622
11-731
11-940
12099
12-758
12-417
12-676
80
12-735
12-895
13 -054
13-213
13-370
13-531
13-690
13-849
14*008
14-168
90
14-327
14-436
14-645
14-804
14-963
15-122
15-281
15-441
15-600
15-759
100
15-918
16-072
16-236
16*395
16-554
16-713
16-873
17 032
17-191
17*350
110
17-504
17-668
17-987
17*827
18*146
18-305
18-464
18-623
18-782
18*941
120
19101
19-260
19-419
19*578
19-737
19-896
20*055
20-214
20-374
20*533
130
20-692
20-851
21 -010
21 *169
21 *328
21*488
21-647
21-806
21*460
22-124
140
22-283
22*442
22-602
22-761
22*920
23-074
23*288
23-397
23-556
23-716
ISO
23 -875
24-034
24-193
24*352
24*511
24-620
24*830
24-989
25-148
25-307
160
25-466
25*625
25-784
25*944
26*103
26-262
26-421
26-580
26-739
26-894
170
27-058
27-217
27-376
27-535
27*694
27-853
27*931
28-172
28-331
28-490
180
28-699
28-808
28-967
29-126
29-286
29-445
29-604
29-763
29-922
30*086
190
30-241
30-400
30*559
30-718
30-877
31-036
31-196
31 355
31-514
31 -673
200
31-832
31-992
32-150
32-310
32-469
S2-628
32-787
32-846
33-105
33*264
210
33-424
33-583
33-742
33-901
34 *060
34*219
34-278
34-537
34-697
34-856
220
35*015
35-174
35-333
35-492
35-652
35-811
35-970
36-129
36-288
36 447
230
36-607
36-766
36-925
37-084
37*243
37-402
37-561
37-720
37-880
38-039
240
38-198
38-357
38-516
38*725
38-835
38-994
39-153
39-312
39-471
39-631
250
39-790
39-949
40-108
40*262
40-426
40-585
40-744
40-904
41*063
41 -222
260
41 -381
41-541
41-699
41 -858
42 019
42*177
42-336
42*495
42*654
42-813
270
42-973
43-132
43*291
43*450
43-609
43*768
43-927
44-087
44-231
44-405
280
44 -564
44-723
44-882
45-042
45-201
45-360
45-519
45-678
45-837
45-996
290
46-156
46-315
46-474
46-633
46*792
46-751
47*111
47-270
47-429
47-588