5

Screw Thread and Gear MeasurementTerminology:

Fig 5. 1Screw thread: a screw thread is the helical ridge produced by forming a continuous helical groove of uniform section on the external or internal surface of a cylinder or a cone. A screw thread formed on a cylinder is known as straight or parallel screw thread, while the one formed on a cone is known as tapered threads.External thread: a thread formed on outside of a work piece is known as external thread. Example: on bolts or studs etc.Internal thread: a thread formed on inside of a work piece is known as internal thread. Example: on a nut or female screw gauge. Multiple-start screw thread: forming two produces this or more helical grooves equally spaced and similarly formed in an axial section on a cylinder. This gives quick traverse without sacrificing core length. Axis of a thread: this is imaginary line running longitudinally through the center of the screw. Hand (right or left hand thread): Suppose a screw is held such that the observer is looking along the axis, if a point moves along the thread in clockwise direction and thus moves away from the observer, the thread is right hand: and if it moves towards the observer the thread is left hand. Form of thread: this is the shape of the contour of one complete thread as seen in axial section. Crest of thread: this is defined as the prominent part of thread, whether it is external or internal. Root of thread: this is defined as bottom of the groove between the two flanks of the thread, whether it is external or internal.Flanks of thread: these are straight edges, which connect the crest with the root. Angle of thread (included angle): this is the angle between the flanks and slope of the thread measured in an axial plane. Flank angle: the flank angles are angles between individual flanks and the perpendicular to the axis of the thread which passes through the vertex of the fundamental angle. The flank angle of a symmetrical thread is commonly termed as the half angle of thread. Pitch: the pitch of the thread is the distance, measured parallel to the axis of the thread, between corresponding points on the adjacent forms in the same axial plane and on the same side of the axis. The basic pitch is equal to the lead divided by the number of the thread starts. On drawings of thread sections, the pitch is shown as the distance from the center of one thread crest to the center of next, and this representation is correct for single start as well as multi-start threads. Lead: lead is the axial distance moved by the threaded part when it is given one complete revolution about its axis with respect to fixed mating thread. the uniformity of pitch measurement does not necessarily assure uniformity of lead. variations in either or pitch cause the functional or virtual diameter of thread to differ from the pitch diameter. Thread per inch: this is the reciprocal of pitch in inches.Lead angle: on straight threads, lead angle is the angle made by the helix of the thread at the pitch line with plane perpendicular to the axis. The angle is measured in actual plane. Helix angle: on a straight thread, the helix angle is the angle made by the helix of the thread at the pitch line with the axis. the angle is measured in an axial plane. Depth of thread: this is the distance from the crest or tip of the thread to the root of the thread-measured perpendicular to the longitudinal axis. This could also be defined as the distance measured radially between the major and minor cylinders. Axially thickness: this is the distance between the opposite faces of the same thread measured on the pitch cylinder in the direction parallel to the axis of the thread.Truncation: a thread is sometimes truncated at the crest or at the root or at both crest and root. Truncation at crest is the radial distance from the crest to nearest apex of the fundamental triangle. Similarly the truncation at the root is the radial distance from the root to the nearest apex. Addendum: for an external thread, this is defined as the radial distance between the major and pitch cylinders. For an internal thread this is the radial distance between the minor and pitch cylinders. Dedendum: this is radial distance between the pitch and minor cylinder for an external thread and for internal thread, this is radial distance between the major and pitch cylinders.Major diameter: in case of a straight thread, this is the diameter of the major cylinder (imaginary cylinder, coaxial with the cylinder, which just touches the roots of an internal thread). It is often referred to as root diameter or cone diameter of external threads.Effective diameter or pitch diameter: in case of straight thread, this is the diameter of the pitch cylinder (the imaginary cylinder which is coaxial with the axis of the screw and intersects the flank of the threads in such a way as to make the width of the threads and width of the spaces between the threads equal.). If the pitch cylinder were imagined as generated by the straight line parallel to the axis of the screw that straight line is referred to as pitch line. Along the pitch line the widths of the threads and the widths of the spaces are equal on a perfect thread. This is the most important dimension as it decides the quality of the fit between screw and nut.Functional (virtual) diameter: for an external or internal thread, this is the pitch diameter of the enveloping thread of perfect pitch, lead and flank angles having full depth of engagement but clear at crest and root. This is defined over a specified length of thread. This may be greater than the effective diameter by an amount due to errors in pitch and angle of thread. The virtual diameter being the modified effective diameter by pitch and angle errors is the most important single dimension of a screw thread gauge. In case of a taper screw thread, the cone angle of taper, for measurement of effective diameter and whether the pitch is measured along the axis or along the pitch code generator also needs to be specified. Errors in threads: In case of plain shafts and holes, there is only one dimension, which has to be considered, and errors on this dimension if exceed the permissible tolerance, will justify the rejection of the part. While in case of screw threads there are at least five important elements, which require consideration, and error in any one of these can cause rejection of the thread. In routine production all of these elements (major dia, minor dia, effective dia, pitch and angle of thread form) must be checked and method of gauging must be able to cover all these elements. Errors on the major and minor diameters will cause interference with the mating thread. Due to errors in these elements, the root section and wall thickness will be less, also the flank contact will be reduced and ultimately the component will be weak in strength. Errors on the effective diameter will also result in weakening of the assembly due to interference between the flanks. Similarly pitch and angle errors are also not desirable as they cause progressive tightening and interference on assembly. These two errors have a special significance as they can be precisely related to effective diameter. Pitch errors in screw threads: A point cutting tool generates Generally screw threads. In this case, for pitch to be correct, the ratio of linear velocity of tool and angular velocity of work must be correct. This ratio must be maintained constant; otherwise pitch errors will occur. If there is any error in the pitch the total length of thread engaged would be either too great or too small, the total pitch in overall length of the thread being called the cumulative pitch error. Various pitch errors are: Progressive pitch errorPeriodic pitch errorDrunken errorIrregular errorsDrunken error: this is the one having erratic pitch, in which the advance of the helix is irregular in one complete revolution of the thread. Thread drunkenness is a particular case of a periodic pitch error recurring at intervals of one pitch. In such a thread, the pitch measured parallel to the pitch measured parallel to the thread axis will always be correct, the error being that the thread is not cut to the true helix. If the screw thread be regarded as an inclined plane wound around the cylinder and if the thread be unwound from the cylinder, (that is development of the thread be taken) then the drunkenness can be visualized. The helix will be a curve in the case of drunken thread and not a straight line as shown in the figure.

Fig 5.2.It is very difficult to determine such errors and moreover they do not have any great effect on the working unless the thread is of very large size. Progressive pitch error: this error occurs when the tool work velocity ratio is incorrect, though it may be constant. It can also be caused due to pitch errors in the lead screw of the lathe or other generating machine. The other possibility is by using an incorrect gear or an approximate gear train between the work and lead screw. E.g. while metric threads are cut with an inch pitch lead screw and a translatory gear are not available. A graph between the cumulative pitch error and the length of thread is generally a straight line in case of progressive error. Periodic pitch error: this repeats itself at regular intervals along the thread. In this case, successive portions of the thread are either shorter or longer than the mean. This type of error occurs when the tool work velocity ratio is not constant. This type of error also results when the thread is cut from a leads crew, which lacks square ness in the abutment causing the leads crew to move back and forth in each revolution. Thus the errors due to these cases are cyclic in nature and so the pitch increases to a maximum value, decreases to the mean and then to the minimum value and so on. The graph between the cumulative pitch error and length of threads for this error will, therefore, be of sinusoidal form. Irregular errors: these arise from the disturbances in the machining setup, variations in the cutting properties of material etc. thus they have no specific causes and correspondingly no specific characteristics also. These errors could be summarized as follows: Erratic pitch: this is irregular error in pitch and varies irregularly in magnitude over different lengths of thread. Progressive error: when the pitch of a screw is uniform, but is shorter or longer than its nominal value, it is said to have progressive error. Periodic error: if the errors vary in magnitude and recur at regular intervals, when measured from thread to thread along the screw are referred to as periodic errors.Screw threads measurements:There are a large number of different standard forms of screw threads in common use. A few important measuring types of screw thread elements are discussed here. Here the nomenclature of the screw threads is not discussed here.Full diameter: for measuring the full diameter of a screw, an ordinary micrometer with anvils of a diameter sufficient to span two threads may be used. To eliminate the effect of errors in the micrometer screw and the measuring faces, it is advisable first to check the instrument on a cylindrical standard of about the same diameter as the screw. For such purposes a plug gauge is useful. Core diameter: the diameter over the root of a thread may be checked by means of a special micrometer adapted with shaped anvils, or an ordinary micrometer may be used in conjunction with a pair of vee pieces. The second method is more universal in application, and a diagram showing the arrangement is given in the figure. It is important that while making the test the micrometer is positioned at right angles to the axis of the screw being measured. The vee pieces used for this test are of hardened steel with an angle of about 450 finished with a radius less than that of the root of the thread. The back faces should be finished flat, perpendicular with the axis of the vee and parallel with the edge of the radius. Effective diameter: the only reliable means of inspecting the effective diameter of a screw is to use some method, which enables a reading to taken from the straight, sloping flanks of the threads. This is accomplished in a simple manner by using small cylindrical test wires, which rest in the thread angle and make contact with the sloping sides. If means are available (e.g. a floating micrometer) for maintaining the micrometer at the right angles to the screw axis, two opposite wires may be used; or else three wires are required to align the micrometer, and this method is the rule when using an ordinary micrometer. The wires should be hardened and polished and their surfaces should be round, straight, parallel, and uniform to a high degree of accuracy. Three-wire method: checking the effective diameter when a screw is measured over wires is given below for general case. One side of the screw is shown in the figure, where w= distance over the wires and DE the effective diameter. The wire is designated with radius r and diameter d.From this general formula we may apply the special adaptation for common threads.

Fig 5.4Pitch:An error in the pitch requires a compensating reduction in effective diameter of approximately twice the amount; pitch errors are to be reduced to absolute minimum. A pitch-measuring device consists of a bed with centers at each end to support the screw, with alternative means for holding nuts and sleeves when internal threads are to be tested. Sliding along the bed and moved by an accurate micrometer is head which carries a feeler piece or stylus shaped to fit in the vee of the thread provided with an indicator which shows when it is bedded home centrally in the vee (i.e. in its lowest position). When making a test, the head is moved along causing the stylus to seat itself successively in each of the threads over the length being examined. Observation and analysis of the micrometer reading obtained then enables the pitch of the thread to be determined. A diagrammatic sketch of the stylus is shown in the figure.

Fig 5.5With a good projection measuring the image and dividing by the magnification may determine the pitch of the portion of the thread. Greater accuracy is obtained if, the measurement is made perpendicular to the thread flanks (instead of measuring parallel to the screw axis), and the result divided by the cosine of half of the thread angle. Thus in figure length AB is measured when pitch AC=AB/cosa . Measurement of gear teeth elements:A few types of measuring gear teeth elements are discussed here. The nomenclature of a toothed is a prerequisite for the following section. The tooth Venire:

Fig 5.6A gear tooth Vernier, figure is provided with two mutually perpendicular scales 1 and 5; the first is used in adjusting for a chordal height and the second, to measure the chordal tooth thickness. Before measurement, the adjustable tongue 3 is set by means of Vernier 2 to the height at which the chordal thickness is to be measured and locked in position. The measuring jaws are moved apart, and after testing the instrument with the tongue on the tip circle of gear being measured, the jaws are drawn closer together and brought into contact with the tooth flanks. The values of the measured chordal thickness are directly read from Vernier 4. Measurement at the constant- chord tooth thickness is preferable (the constant chord is the chord between the points of contact of the basic rack profile with the tooth flanks at a normal section). The nominal values of the constant chord height and tooth thickness are selected from the corresponding tables compiled or are calculated by the corresponding formulae. For standard spur gears with a normal pressure angle of 200< the constant-chord height h equal to h=0.7476m. And the constant chord tooth thickness is S=1.387m. Where m is module, mm. Base pitch:The base pitch is the circular pitch of the teeth measures on the base circle. The tooth span micrometer is used to check the mean value and variation in the base tangent length. It varies from the standard micrometers only with respect to the measuring anvils. Here disk type measuring anvils are used. The disk anvil frame may be partly cut away. These micrometers are often used to determine an unknown gear module. To this end the base tangent length is measured first over n teeth then over n-1 teeth. The difference in measurement gives the base pitch t0 which is used for module by the formula m=t0/p cosf where f is the pressure angle. Gear MeasurementsThe most commonly used forms of gear teeth are involute & cycloid. The involute tooth is derived from the trace of the point on a straight line, which rolls without slipping around a circle, which is the base circle, or it could be defined as a locus of a point on a piece of string which is unwounded from a stationary cylinder. The cycloid tooth is derived from the curve, which is the locus of a point on a circle rolling on the pitch circle of the gear. Here the addendum tooth is the trace of the point on a circle rolling outside of the pitch circle and this is an epicycloidal curve whereas the dedendum portion of the tooth is the trace of the point on a circle rolling on the inside of the pitch circle of the gear and is hypocycloidal gear.

The various types of commonly used gears are:

Spur gear: it is a cycloid gear whose tooth traces is straight line.

Helical gear: it is a cylindrical gear whose tooth traces is straight helices.

Spiral gear: a gear whose tooth traces is curved line.

Straight bevel gear: a gear whose tooth traces is a straight-line generator of a cone. It is conical in form in operating and intersecting axes usually at angles.

Worm gear pair: the worm and mating worm wheel have their axes non-parallel and non-intersecting.

Gear Terminologies

FIG.5.7

PITCH CIRCLEWhen two gears are meshed and running there are two circles which appear to roll one on another. These two rolling circles are called pitch circles. Diameter of the gear is represented by diameter of the pitch circles and is denoted by "d".

ADDENDUM CIRCLEIt is a circle, which passes through the tip of the tooth.

DEDENDUM CIRCLEIt is a circle, which passes through the root of the tooth.

TOOTH THICKNESSIt is the thickness of the tooth measured along the pitch circle.

SPACE WIDTHIt is the distance between two adjacent teeth measured along the pitch circle.

CIRCULAR PITCH (P or Pc)It is the distance from a point on one tooth to a similar point on the adjacent tooth measured along the pitch circle. It is also the ratio of the circumference of the pitch circle to the number of teeth.

Pc = d/t

Where t number of teeth

FACE WIDTHIt is the length of the tooth measured parallel to the axis of the gear.

ADDENDUMIt is the radial height of the tooth between the pitch circle and addendum circle.

DEDENDUMIt is the radial height of the tooth between the pitch circle and dedendum circle.

FACEIt is the working area of the tooth between addendum circle and pitch circle.

FLANKIt is the working area of the tooth between pitch circle and dedendum circle.

MODULE (m)It is the diameter measured per tooth of the gear. It is always represented in mm only m= d/t

But Pc = d/t

Pc = m

DIAMETRAL PITCH (Pd)It is a reciprocal of module of the number of teeth per mm of diameter.

PITCH POINTIt is the point of contact or tangency of two pitch circles.

LINE OF CONTACTIt is the line along which the points of contact between two pairs of teeth proceed.

PRESSURE ANGLEIt is the angle between the line of contact and the common tangent at the pitch point.

CLEARANCEIt is the difference between the dedendum and addendum.

BACKLASHIt is the difference between the space width and tooth thickness.

LENGTH OF PATH OF CONTACTIt is the distance measured along the line of contact from the point of engagement to the point of disengagement.

GEAR RATIO (G)It is the ratio of the gear diameter to the pinion diameter or the ratio of the pinion speed to the gear speed or ratio of number of teeth on gear to that on pinion.

G = D/d = n/N = T/t

Measurement of individual elementsMeasurement of tooth thicknessThe permissible error or the tolerance on thickness of tooth is the variation of actual thickness of tooth from its theoretical value the tooth thickness is generally measured at pitch circle and is therefore, the pitch line thickness of the tooth. It may be mentioned that the tooth thickness is defined as the length of an arc, which is difficult to measure directly. In most of the cases, it is sufficient to measure the chordal thickness that is the cord joining the intersection of the tooth profile with the pitch circle. Also the difference between chordal tooth thickness and circular tooth thickness is very small for gear of small pitch. The thickness measurement is the most important measurement because most of the gears manufactured may not undergo checking of all other parameters, but thickness measurement is a must for all gears. There are various methods of measuring the gear tooth thickness:

Measurement of tooth thickness byGear tooth vernier caliper.

Constant chord method.

Base tangent method.

Measurement by dimension over pins

The tooth thickness can be very conveniently measured by a gear tooth vernier. Since the tooth thickness varies from the tip of the base circle of the tooth, the instrument must be capable of measuring the tooth thickness at a specified position on the tooth. Further this is possible only when there is some arrangement to fix that position where the measurement is to be taken. The tooth thickness is generally measured at pitch circle & is, therefore, referred to as pitch line thickness of tooth. The gear tooth in the vernier has two vernier scales & they are set for the width w of the tooth & the depth d from the top, at which w occurs.

FIG- 5.8Considering one gear tooth, the theoretical values of w & d can be found out which may be verified by the instrument. In the fig it may be noted that w is a chord ADB, but tooth thickness specified as an arc distance AEB. Also the distance d adjusted on instrument is slightly greater than the addendum CE, w is therefore called chordal thickness & d is called the chordal addendum.

From the fig, w=AB=2AD,

Now angle AOD = = 3600/4N

Where N is the number of teeth,

w=2AD=2*AO*sin= 2R sin (360/4N) (R=PITCH CIRCLE RADIUS)

Module, m= P.C.D/number of teeth = 2R/N

R=N*m/2

w=(N*m)*sin(360/4N)

Also from fig, d= OC-OD

OC = OE+ addendum = R+m

= (N*m/2)+m

OD = R * cos= N*m/2 cos(90/N)

d = (N*m/2)+m-(N*m/2) cos(90/N)

Any error in the outside diameter of the gear must be allowed for when measuring tooth thickness.

In case of helical gears the above expressions must have to be modified to take into account the change in curvature along the pitch line. These formulae apply when backlash is ignored.

Gear tooth Caliper

FIG-5.9

It is used to measure the thickness of gear teeth at the pitch line or chordal thickness of teeth & the distance from the top of a tooth to the chord. An adjustable tongue, each of which is adjusted independently by adjusting the screw on graduated bars, measures the thickness of the tooth at pitch line & the addendum. The effect of zero errors should be taken into consideration.

This method is simple & inexpensive. However it needs different setting for a variation in number of teeth for a given pitch & accuracy is limited by the least count of instrument. Since the wear during use is concentrated on the two jaws, caliper has to be calibrated at regular intervals to maintain the accuracy of measurement.

Gear tooth VernierMost of the times a gear Vernier is used to measure the tooth thickness. As the tooth thickness varies from top to the bottom, any instrument for measuring on a single tooth must.

Fig 5.10A gear tooth Vernier, figure is provided with two mutually perpendicular scales 1 and 5; the first is used in adjusting for a chordal height and the second, to measure the chordal tooth thickness. Before measurement, the adjustable tongue 3 is set by means of Vernier 2 to the height at which the chordal thickness is to be measured and locked in position. The measuring jaws are moved apart, and after testing the instrument with the tongue on the tip circle of gear being measured, the jaws are drawn closer together and brought into contact with the tooth flanks.

The values of the measured chordal thickness are directly read from Vernier 4. Measurement at the constant- chord tooth thickness is preferable (the constant chord is the chord between the points of contact of the basic rack profile with the tooth flanks at a normal section). The nominal values of the constant chord height and tooth thickness are selected from the corresponding tables compiled or are calculated by the corresponding formulae.

For standard spur gears with a normal pressure angle of 200< the constant-chord height h equal to h=0.7476m.

And the constant chord tooth thickness is

S=1.387m.

Where m is module, mm.

Base pitchThe base pitch is the circular pitch of the teeth measures on the base circle. The tooth span micrometer is used to check the mean value and variation in the base tangent length. It varies from the standard micrometers only with respect to the measuring anvils. Here disk type measuring anvils are used. The disk anvil frame may be partly cut away. These micrometers are often used to determine an unknown gear module. To this end the base tangent length is measured first over n teeth then over n-1 teeth. The difference in measurement gives the base pitch t0 which is used for module by the formula m=t0/ cos where is the pressure angle.

FIG:5.11