Modern Machine-Shop Practice Part 17

The following table gives the safe working pressures for wheels having an inch pitch and an inch face when working at the given velocities, S.W.P. standing for "safe working pressure:"--

+------------+------------+------------+------------+--------------+ | Velocity of| | | | | |pitch circle| S.W.P. for | S.W.P. for |S.W.P. for | S.W.P. | | in feet | cast-iron |spur mortise| cast-iron | for bevel | |per second. |spur gears. | gears. |bevel gears.|mortise gears.| +------------+------------+------------+------------+--------------+ | 2 | 368 | 178 | 258 | 178 | | 3 | 322 | 178 | 225 | 157 | | 6 | 255 | 178 | 178 | 125 | | 12 | 203 | 142 | 142 | 99 | | 18 | 177 | 124 | 124 | 87 | | 24 | 161 | 113 | 113 | 79 | | 30 | 150 | 105 | 105 | 74 | | 36 | 140 | 98 | 98 | 69 | | 42 | 133 | 93 | 93 | 65 | | 48 | 127 | 88 | 88 | 62 | +------------+------------+------------+------------+--------------+

For velocities less than 2 feet per second, use the same value as for 2 feet per second.

The proportions, in terms of the pitch, upon which this table is based, are as follows:--

Thickness of iron teeth .395 of the pitch.

" wooden " .595 "

Height of addendum .28 "

Depth below pitch line .32 "

The table is based upon 400 lbs. per inch of face for an inch pitch, as the safe working pressure of mortise wheel teeth or cogs; it may be noted that there is considerable difference of opinion. They are claimed by some to be in many cases practically stronger than teeth of cast iron. This may be, and probably is, the case when the conditions are such that the teeth being rigid and rigidly held (as in the case of cast-iron teeth), there is but one tooth on each wheel in contact. But when there is so nearly contact between two teeth on each wheel that but little elasticity in the teeth would cause a second pair of teeth to have contact, then the elasticity of the wood would cause this second contact. Added to this, however, we have the fact that under conditions where violent shock occurs the cog would have sufficient elasticity to give, or spring, and thus break the shock which cast iron would resist to the point of rupture. It is under these conditions, which mainly occur in high velocities with one of the wheels having cast teeth, that mortise wheels, or cogging, is employed, possessing the advantage that a broken or worn-out tooth, or teeth, may be readily replaced. It is usual, however, to a.s.sign to wooden teeth a value of strength more nearly equal to that of its strength in proportion to that of cast iron; hence, Thomas Box allows a wood tooth a value of about 3/10ths the strength of cast iron; a value as high as 7/10ths is, however, a.s.signed by other authorities. But the strength of the tooth cannot exceed that at the top of the shank, where it fits into the mortise of the wheel, and on account of the leverage of the pressure the width of the mortise should exceed the thickness of the tooth.

In some practice, the mortise teeth, or cogs, are made thicker in proportion to the pitch than the teeth on the iron wheel; thus Professor Unwin, in his "Elements of Machine Design," gives the following as "good proportions":--

Thickness of iron teeth 0.395 of the pitch.

" wood cogs 0.595 " "

which makes the cogs 2/10ths inch thicker than the teeth.

The mortises in the wheel rim are made taper in both the breadth and the width, which enables the tooth shank to be more accurately fitted, and also of being driven more tightly home, than if parallel. The amount of this taper is a matter of judgment, but it may be observed that the greater the taper the more labor there is involved in fitting, and the more strain there is thrown upon the pins when locking the teeth with a given amount of strain. While the less the taper, the more care required to obtain an accurate fit. Taking these two elements into consideration, 1/8th inch of taper in a length of 4 inches may be given as a desirable proportion.

[Ill.u.s.tration: Fig. 182.]

As an evidence of the durability of wooden teeth, there appeared in _Engineering_ of January 7th, 1879, the ill.u.s.tration shown in Fig. 182, which represents a cog from a wheel of 14 ft. 1/2 in. diameter, and having a 10-inch face, its pinion being 4 ft. in diameter. This cog had been running for 26-1/2 years, day and night; not a cog in the wheel having been touched during that time. Its average revolutions were 38 per minute, the power developed by the engine being from 90 to 100 indicated horse-power. The teeth were composed of beech, and had been greased twice a week, with tallow and plumbago ore.

Since the width of the face of a wheel influences its wear (by providing a larger area of contact over which the pressure may be distributed, as well as increasing the strength), two methods of proportioning the breadth may be adopted. First, it may be made a certain proportion of the pitch; and secondly, it may be proportioned to the pressure transmitted and the number of revolutions. The desirability of the second is manifest when we consider that each tooth will pa.s.s through the arcs of contact (and thus be subjected to wear) once during each revolution; hence, by making the number of revolutions an element in the calculation to find the breadth, the latter is more in proportion to the wear than it would be if proportioned to the pitch.

It is obvious that the breadth should be sufficient to afford the required degree of strength with a suitable factor of safety, and allowance for wear of the smallest wheel in the pair or set, as the case may be.

According to Reuleaux, the face of a wheel should never be less than that obtained by multiplying the gross pressure, transmitted in lbs., by the revolutions per minute, and dividing the product by 28,000.

In the case of bevel-wheels the pitch increases, as the perimeter of the wheel is approached, and the maximum pitch is usually taken as the designated pitch of the wheel. But the mean pitch is that which should be taken for the purposes of calculating the strength, it being in the middle of the tooth breadth. The mean pitch is also the diameter of the pitch circle, used for ascertaining the velocity of the wheel as an element in calculating the safe pressure, or the amount of power the wheel is capable of transmitting, and it is upon this basis that the values for bevel-wheels in the above table are computed.

In many cases it is required to find the amount of horse-power a wheel will transmit, or the proportions requisite for a wheel to transmit a given horse-power; and as an aid to the necessary calculations, the following table is given of the amount of horse-power that may be transmitted with safety, by the various wheels at the given velocities, with a wheel of an inch pitch and an inch face, from which that for other pitches and faces may be obtained by proportion.

TABLE SHOWING THE HORSE-POWER WHICH DIFFERENT KINDS OF GEAR-WHEELS OF ONE INCH PITCH AND ONE INCH FACE WILL SAFELY TRANSMIT AT VARIOUS VELOCITIES OF PITCH CIRCLE.

+------------+------------+------------+-------------+-------------+ |Velocity of | | | | | |Pitch Circle|Spur-Wheels.|Spur Mortise|Bevel-Wheels.|Bevel Mortise| |in Feet per | H.P. | Wheels. | H.P. | Wheels. | |Second. | | H.P. | | H.P. | +------------+------------+------------+-------------+-------------+ | 2 | 1.338 | .647 | .938 | .647 | | 3 | 1.756 | .971 | 1.227 | .856 | | 6 | 2.782 | 1.76 | 1.76 | 1.363 | | 12 | 4.43 | 3.1 | 3.1 | 2.16 | | 18 | 5.793 | 4.058 | 4.058 | 2.847 | | 24 | 7.025 | 4.931 | 4.931 | 3.447 | | 30 | 8.182 | 5.727 | 5.727 | 4.036 | | 36 | 9.163 | 6.414 | 6.414 | 4.516 | | 42 | 10.156 | 7.102 | 7.102 | 4.963 | | 48 | 11.083 | 7.680 | 7.680 | 5.411 | +------------+------------+------------+-------------+-------------+

In this table, as in the preceding one, the safe working pressure for 1-inch pitch and 1-inch breadth of face is supposed to be 400 lbs.

In cast gearing, the mould for which is made by a gear moulding machine, the element of draft to permit the extraction of the pattern is reduced: hence, the pressure of tooth upon tooth may be supposed to be along the full breadth of the tooth instead of at one corner only, as in the case of pattern-moulded teeth. But from the inaccuracies which may occur from unequal contraction in the cooling of the casting, and from possible warping of the casting while cooling, which is sure to occur to some extent, however small the amount may be, it is not to be presumed that the contact of the teeth of one wheel will be in all the teeth as perfect across the full breadth as in the case of machine-cut teeth.

Furthermore, the clearance allowed for machine-moulded teeth, while considerably less than that allowed for pattern-moulded teeth, is greater than that allowed for machine-cut teeth; hence, the strength of machine-moulded teeth in proportion to the pitch lies somewhere between that of pattern-moulded and machine-cut teeth--but exactly where, it would be difficult to determine in the absence of experiments made for the purpose of ascertaining.

It is not improbable, however, that the contact of tooth upon tooth extends in cast gears across at least two-thirds of the breadth of the tooth, in which case the rules for ascertaining the strength of cut teeth of equal thickness may be employed, subst.i.tuting 2/3rds of the actual tooth breadth as the breadth for the purposes of the calculation.

If instead of supposing all the strain to fall upon one tooth and calculating the necessary strength of the teeth upon that basis (as is necessary in interchangeable gearing, because these conditions may exist in the case of the smallest pinion that can be used in pitch), the actual working condition of each separate application of gears be considered, it will appear that with a given diameter of pitch circle, all other things being equal, the arc of contact will remain constant whatever the pitch of the teeth, or in other words is independent of the pitch, and it follows that when the thickness of iron necessary to withstand (with the allowances for wear and factor of safety) the given stress under the given velocity has been determined, it may be disposed in a coa.r.s.e pitch that will give one tooth always in contact, or a finer pitch that will give two or more teeth always in contact, the strength in proportion to the duty remaining the same in both cases.

In this case the expense of producing the wheel patterns or in tr.i.m.m.i.n.g the teeth is to be considered, because if there are a train of wheels the finer pitch would obviously involve the construction and dressing to shape of a much greater number of teeth on each wheel in the train, thus increasing the labor. When, however, it is required to reduce the pinion to a minimum diameter, it is obvious that this may be accomplished by selecting the finer pitch, because the finer the pitch, the less the diameter of the wheel may be. Thus with a given diameter of pitch circle it is possible to select a pitch so fine that motion from one wheel may be communicated to another, whatever the diameter of the pitch circle may be, the limit being bounded by the practicability of casting or producing teeth of the necessary fineness of pitch. The durability of a wheel having a fine pitch is greater for two reasons: first, because the metal nearest the cast surface of cast iron is stronger than the internal metal, and the finer pitch would have more of this surface to withstand the wear; and second, because in a wheel of a given width there would be two points, or twice the area of metal, to withstand the abrasion, it being remembered that the point of contact is a line which partly rolls and partly slides along the depth of the tooth as the wheel rotates, and that with two teeth in contact on each wheel there are two of such lines. There is also less sliding or rubbing action of the teeth, but this is offset by the fact that there are more teeth in contact, and that there are therefore a greater number of teeth simultaneously rubbing or sliding one upon the other.

But when we deal with the number of teeth the circ.u.mstances are altered; thus with teeth of epicycloidal form it is manifestly impossible to communicate constant motion with a driving wheel having but one tooth, or to receive motion on a follower having but one tooth. The number of teeth must always be such that there is at all times a tooth of each wheel within the arc of action, or in contact, so that one pair of teeth may come into contact before the contact of the preceding teeth has ceased.

In the construction of wheels designed to transmit power as well as simple motion, as is the case with the wheels employed in machine work, however, it is not considered desirable to employ wheels containing a less number of teeth than 12. The diameter of the wheel bearing such a relation to the pitch that both wheels containing the same number of teeth (12), the motion will be communicated from one to the other continuously.

It is obvious that as the number of teeth in one of the wheels (of a pair in gear) is increased the number of teeth in the other may be (within certain limits) diminished, and still be capable of transmitting continuous motion. Thus a pinion containing, say 8 teeth, may be capable of receiving continuous motion from a rack in continuous motion, while it would not be capable of receiving continuous motion from a pinion having 4 teeth; and as the requirements of machine construction often call for the transmission of motion from one pinion to another of equal diameters, and as small as possible, 12 teeth are the smallest number it is considered desirable for a pinion to contain, except it be in the case of an internal wheel, in which the arc of contact is greater in proportion to the diameters than in spur-wheels, and continuous motion can therefore be transmitted either with coa.r.s.er pitches or smaller diameters of pinion.

For convenience in calculating the pitch diameter at pitch circle, or pitch diameter as it is termed, and the number of teeth of wheels, the following rules and table extracted from the _Cincinnati Artisan_ and arranged from a table by D. A. Clarke, are given. The first column gives the pitch, the following nine columns give the pitch diameters of wheels for each pitch from 1 tooth to 9. By multiplying these numbers by 10 we have the pitch diameters from 10 to 90 teeth, increasing by _tens_; by multiplying by 100 we likewise have the pitch diameters from 100 to 900, increasing by _hundreds_.

TABLE FOR DETERMINING THE RELATION BETWEEN PITCH DIAMETER, PITCH, AND NUMBER OF TEETH IN GEAR-WHEELS.

+-----+------------------------------------------------------------------+ | | NUMBER OF TEETH. | |Pitch.------+------+------+------+------+-------+-------+-------+-------+ | | 1. | 2. | 3. | 4. | 5. | 6. | 7. | 8. | 9. | +-----+------+------+------+------+------+-------+-------+-------+-------+ |1 | .3183| .6366| .9549|1.2732|1.5915| 1.9099| 2.2282| 2.5465| 2.8648| |1-1/8| .3581| .7162|1.0743|1.4324|1.7905| 2.1486| 2.5067| 2.8648| 3.2229| |1-1/4| .3979| .7958|1.1937|1.5915|1.9894| 2.3873| 2.7852| 3.1831| 3.5810| |1-3/8| .4377| .8753|1.3130|1.7507|2.1884| 2.6260| 3.0637| 3.5014| 3.9391| | | | | | | | | | | | |1-1/2| .4775| .9549|1.4324|1.9099|2.3873| 2.8648| 3.3422| 3.8197| 4.2971| |1-5/8| .5173|1.0345|1.5517|2.0690|2.5862| 3.1035| 3.6207| 4.1380| 4.6552| |1-3/4| .5570|1.1141|1.6711|2.2282|2.7852| 3.3422| 3.8993| 4.4563| 5.0134| |1-7/8| .5968|1.1937|1.7905|2.3873|2.9841| 3.5810| 4.1778| 4.7746| 5.3714| | | | | | | | | | | | |2 | .6366|1.2732|1.9099|2.5465|3.1831| 3.8197| 4.4563| 5.0929| 5.7296| |2-1/8| .6764|1.3528|2.0292|2.7056|3.3820| 4.0584| 4.7348| 5.4112| 6.0877| |2-1/4| .7162|1.4324|2.1486|2.8648|3.5810| 4.2972| 5.0134| 5.7296| 6.4457| |2-3/8| .7560|1.5120|2.2679|3.0239|3.7799| 4.5359| 5.2919| 6.0479| 6.8038| | | | | | | | | | | | |2-1/2| .7958|1.5915|2.3873|3.1831|3.9789| 4.7746| 5.5704| 6.3662| 7.1619| |2-5/8| .8355|1.6711|2.5067|3.3422|4.1778| 5.0133| 5.8499| 6.6845| 7.5200| |2-3/4| .8753|1.7507|2.6260|3.5014|4.3767| 5.2521| 6.1274| 7.0028| 7.8781| |2-7/8| .9151|1.8303|2.7454|3.6605|4.5757| 5.4908| 6.4059| 7.3211| 8.2362| | | | | | | | | | | | |3 | .9549|1.9099|2.8648|3.8197|4.7746| 5.7296| 6.6845| 7.6394| 8.5943| |3-1/4|1.0345|2.0690|3.1035|4.1380|5.1725| 6.2070| 7.2415| 8.2760| 9.3105| |3-1/2|1.1141|2.2282|3.3422|4.4563|5.5704| 6.6845| 7.7986| 8.9126|10.0268| |3-3/4|1.1937|2.3873|3.5810|4.7746|5.9683| 7.1619| 8.3556| 9.5493|10.7429| | | | | | | | | | | | |4 |1.2732|2.5465|3.8197|5.0929|6.3662| 7.6394| 8.9127|10.1839|11.4591| |4-1/2|1.4324|2.8648|4.2972|5.7296|7.1619| 8.5943|10.0267|11.4591|12.8915| |5 |1.5915|3.1831|4.7746|6.3662|7.9577| 9.5493|11.1408|12.7324|14.3240| |5-1/2|1.7507|3.5014|5.2521|7.0028|8.7535|10.5042|12.2549|14.0056|15.7563| |6 |1.9099|3.8196|5.7295|7.6394|9.5493|11.4591|13.3690|15.2788|17.1887| +-----+------+------+------+------+------+-------+-------+-------+-------+

The following rules and examples show how the table is used:

Rule 1.--Given ---- number of teeth and pitch; to find ---- pitch diameter.

Select from table in columns opposite the given pitch--

First, the value corresponding to the number of units in the number of teeth.

Second, the value corresponding to the number of tens, and multiply this by 10.

Third, the value corresponding to the number of hundreds, and multiply this by 100. Add these together, and their sum is the pitch diameter required.

Example.--What is the pitch diameter of a wheel with 128 teeth, 1-1/2 inches pitch?

We find in line corresponding to 1-1/2 inch pitch--

Pitch diameter for 8 teeth 3.8197 " " 20 " 9.549 " " 100 " 47.75 --- ------- " " 128 " 61.1187

Or about 61-1/8". Answer.

Rule 2.--Given ---- pitch diameter and number of teeth; to find ---- pitch.

First, ascertain by Rule 1 the pitch diameter for a wheel of 1-_inch pitch_, and the given _number of teeth_.

Second, divide _given pitch diameter_ by the _pitch diameter_ for 1-_inch pitch_.

The quotient is the pitch desired.

Example.--What is the pitch of a wheel with 148 teeth, the pitch diameter being 72"?

First, pitch diameter for 148 teeth, 1-inch pitch, is--

8 teeth 2.5465 40 " 12.732 100 " 31.83 --- ------- 148 " 47.1085