Screw Threads & Gear Measurements

Screw Threads Screw threads are used; ‡ To hold parts together (ex: V-threads) ‡ To transmit motion & power (Square, Acme threads) Screw threads Terminology ‡ PITCH: The distance from a point on a screw thread to a corresponding point on the next thread measured parallel to the axis. ‡ LEAD: The distance a screw thread advances in one turn. For a single start threads, lead=pitch, For double start, lead=2xpitch, & so on. ‡ THREAD FORM: The cross section of thread cut by a plane containing the axis. ‡ MAJOR DIAMETER: This is the diameter of an imaginary cylinder, co-axial with the screw, which just touches the crests of an external thread or roots of an internal threads. It is also called as µNominal diameter¶. SCREW THREAD TEMINOLOGY EXTERNAL THREAD TERMINOLOGY Pitch Crest Flank Pitch line Thread Angle Axial thickness Flank angle Addendum Dedendum Root Axis of thread Major dia Pitch dia Minor dia Screw threads Terminology ‡ Minor diameter: This is the diameter of an imaginary cylinder, co-axial with the screw which just touches the roots of an external thread or the crest of an internal thread. This is also referred to as µroot¶ or µcore diameter¶. ‡ Effective diameter or Pitch diameter: It is the diameter of an imaginary cylinder coaxial with the axis of the thread and intersects the flanks of the thread such that width of the threads & width of spaces between threads are equal. ‡ Flank: It is the Thread surface that connects crest with root. ‡ Depth of thread: It is the distance between crest and root measured perpendicular to axis of screw. Screw threads Terminology ‡ Angle of thread: Included angle between sides of thread measured in axial plane. ‡ Helix angle: Angle that thread makes with plane perpendicular to thread axis. ‡ Flank angle: It is half the included angle of the thread. ‡ Addendum: It is the distance between the crest and the pitch line measured perpendicular to axis of the screw. ‡ Dedendum: It is the distance between the pitch line & the root measured perpendicular to axis of the screw. Errors in screw threads ‡ There are six important elements in a thread, errors in any one of which may lead to rejection. ‡ They are Major dia, Minor dia, effective dia, Pitch, Flank angle and the profile at root & crest. ‡ Errors in screw threads may cause interference with mating threads or slackness due to improper flank contact. ‡ Errors in pitch of screw thread may be classified into three types; (i) Periodic errors (ii) Progressive errors (iii) Erratic errors (i) Periodic errors ‡ Periodic errors are those which vary in magnitude along the length of the thread and occurs at regular intervals as shown in fig (a). ‡ A µdrunken thread¶ is a particular case of periodic error where the error repeats once per turn. ‡ For a true thread, if the thread is imagined to be unwound from the pitch cylinder, the helix will be a straight line. For a drunken thread, it will be a curve as shown in fig (b). ‡ In a drunken thread, the advance of the helix is irregular in one complete revolution. This is due to thread being not cut to true helix. (i) Periodic errors + True thread Cumulative pitch error Pitch Max error Drunken thread _ Length of thread TMean diameter Fig (a) PERIODIC ERRORS Fig (b) DRUNKEN THREAD ‡ Progressive pitch error: If the pitch of the thread is uniform but is longer or shorter than its nominal value, then the error is called progressive as shown in fig (c). These errors may be caused by a change in length due to hardening, or by the errors in the pitch of the lead screw, or by the faults in the saddle guide ways. ‡ Erratic errors: These errors vary in irregular manner along the length of the thread as shown in fig (d). Their causes are difficult to identify. Possible sources are faults in the machine and irregular cutting action resulting from material non uniformity. (ii) Progressive & (iii) Erratic errors umulative pitch error umulative pitch error Length o thread Length o thread ig (c) IV ig (d) TI G Measurement of major diameter Clamp iducial Indicator upports easuring nvils olding centres icrometer head B CH ICROMETER Bench Micrometer BENCH MICROMETER Bench Micrometer ‡ A good quality hand held micrometer is quite suitable for measuring external thread, but only light pressure has to be applied on the anvils to make only contact on the screw threads. ‡ Excessive pressure may lead to elastic deformation of screw threads leading to errors. ‡ A bench micrometer may be used for greater accuracy which give direct readings of 0.0002 mm. ‡ A standard cylinder of known diameter µS¶ (which is nearly equal to thread diameter) is held between centers & a reading R1 between the fiducial indicator anvil & micrometer anvil is taken. The cylinder is then removed. ‡ Then the screw thread to be measured is held between centers & a second micrometer reading R2 is taken. ‡ Then D ! S s (R ~ R ) 1 1 2 Holding centre Holding centre Standard Cylinder Measuring anvil Screw Thread Measuring anvil Measurement of Major diameter Measurement of minor diameter ‡ The principle of minor diameter is same as that of measuring major diameter except that v shaped prisms are used. ‡ Prisms of suitable sizes are placed between the standard cylinder and the instrument anvils in order to take a reading first micrometer reading R1 . ‡ The standard cylinder is then replaced by the screw thread and a second reading R2 is taken as shown in fig. ‡ Then the minor diameter D 2 ! S s (R 1 ~ R 2 ) R1 S Holding centre iducial indicator anvil rism Standard Cylinder R2 rism Micrometer anvil Measurement o Minor diameter Scre Thread Measurement of Flank angle 1 0 20 0 1 0 20 rotractor arm rotractor Scre ivot Shado rotractor Shadow protractor ‡ Shadow protractor is most convenient method for measurement of flank angles using optical projection. ‡ The shadow of the thread is viewed on a screen and the angles are measured by means of a protractor. ‡ For clear definition of a thread form on the screen it is necessary to project the light beam along the thread helix angle by using a lamp & collimating unit. ‡ The protractor is supported at the screen on a straight edge. The pivoted arm of the protractor is rotated until its shadow is parallel to the flank & the first reading is taken. ‡ The screw is then rotated 90o to its axis and the protractor is swung about its pivot and adjusted to measure the angle of the same flank and a second reading is taken. ‡ The mean of the two readings is then the angle between the flank & normal to the screw axis. Measurement of effective diameter by Two wire method MEASUREMENT B TWO WIRE METHOD E M Dia 'd' U 3 P Pitch line C B D F E G % C U M 8 E Measurement of effective diameter using two wire method ‡ The effective diameter can not be measured directly but can be calculated from the measurements made. ‡ Wires of exactly known diameters are chosen such that they contact the flanks at their straight portions. ‡ If the size of the wire is such it contacts the flanks at the pitch line, it is called the µbest size¶ of wire which can be determined by geometry of screw thread. ‡ The screw thread is mounted between the centers & wires are placed in the grooves and reading M is taken. ‡ Then the effective diameter E =T+C where T =M-2d, & C is a value which depends on diameter of wire, pitch & angle of the screw thread. Measurement of effective diameter using two wire method U d U From the triangle OAB, OA ! OB.cosec ! . cos ec 2 2 2 d¨ d» U ¸ « AG ! (OA  OG ) ! © cos ec  1¹ ¬3 OG ! OB ! 2 ¼ 2ª 2 º ½ ­ P U In the triangle FD, F ! DF.cot . But DF ! 2 4 U d¨ U ¸ P and FG ! ( F  G) ! . cot  © cos ec  1¹ 4 2 2ª 2 º Effective diameter E ! T  ( 2 v C) here C ! 2 v FG U d¨ U ¸¾ P ® @ Effective diameter E ! T  (2 v FG ) ! T  2¯ . cot  © cos ec  1¹¿ 4 2 2ª 2 ºÀ ° P U ¨ U ¸¾ ® @ E ! T  2¯ . cot  d© cos ec  1¹¿ 2 2 ª 2 ºÀ ° For hit orth thread, U ! 55o @ C ! 0.9605P  1.1657d For metric thread U ! 60o @ C ! 0.866P  d Effective diameter by three wire method E M Dia 'd' P U A pitch line H h B C U D M E Effective diameter by three wire method ‡ This method is more accurate than two wire method as it ensures alignment of micrometer faces parallel to the thread axis. ‡ Here, three wires of exactly known diameters are used, one on one side & the two on the other side. The wires may be held in hand or hung from a stand. ‡ From the fig, M=diameter over the wires E= effective diameter (to be found) d= diameter of wires, h=height of wire center above the pitch line, r=radius of wire, H=depth of thread, D=major diameter of the thread. Effective diameter by three wire method U d U From the triangle A D , AD ! A cosec ! cosec 2 2 2 H P U P U U and CD ! ! cot H ! DE cot ! cot 2 2 2 2 4 2 U » «P U» «d Further h ! ( AD  CD ) ! ¬ cosec ¼  ¬ cot ¼ 2½ ­ 4 2½ ­2 Distance over the wires, M ! E  2h  2r U P U¾ U¾ P U i.e. M ! E  2¯rcosec  cot ¿  2r ! E 2r ¯1 cosec ¿  cot 2 4 2À 2À 2 2 U¾ P U Or M ! E d ¯1 cosec ¿ cot 2À 2 2 For Whitworth thread, U ! 55o , depth of thread ! 0.64P U U @ E ! D - 0.64P, cosec ! 2.1657, and cot ! 1.921 2 2 M ! D  3.1657d  1.605P where D is the major diameter of the thread. For Metric threads, Depth of thread ! 0.6495P U U @ E ! D  0.6495P, U ! 60 , cos ec ! 2, cot ! 1.732 2 2 @ M ! D  3d  1.5155P o We can measure the value of M practically & then compare with the theoretical values using formulae derived above. After finding the correct value of M, as d is known, E can be found out. Expression for Best size Wire P/4 P 3 Pitch line B A U ) P/2 BEST SIZE OF WIRE Expression for Best size Wire The best size wire is the one which makes contact at the pitch line or effective diameter of the screw thread. In other words, as shown in fig O is perpendicular to flank portion of the thread at the pitch line. U¸ A ¨ › ¸ A ¨ , or sin © 90 - ¹ ! In the triangle OA , Sin © O A ¹ ! 2º O ª ª º O A U A @O ! ! ! A sec . U U¸ 2 ¨ cos sin © 90 - ¹ 2 2º ª 1 ut O ! radius of wire ! v dia of best size wire (D b ) 2 P U i.e. D b ! 2 v O ! 2 v A sec . Also since A lies on the pitch line, A ! 4 2 where P is the pitch of the thread. P U P U @ D b ! 2 sec ! sec 4 2 2 2 Pitch Measuring machine Fiducial indicator Base Pointer K Pointer T Spring loaded head stylus Micrometer centers Carriage Screw thread Pitch Measuring machine ‡ For measuring pitch, two methods are commonly employed as follows; (a) Using pitch measuring machine (b) Using Toolmaker¶s microscope ‡ In a pitch measuring machine, the screw thread is mounted between the centers of the machine. A stylus inserted into a spring loaded head makes contact at the thread flanks near the pitch line. ‡ The spring loaded head permits the stylus to move up the flank of the thread & down into the next space as it is moved parallel to the axis. ‡ Accurate positioning of the stylus between the two flanks is ensured by keeping the pointer T is always opposite to its index mark while taking readings. Pitch Measuring machine (contd«) ‡ With the micrometer reading zero on the scale, the indicator is moved along to bring the stylus opposite to the first thread space and is clamped in position. ‡ The indicator is then adjusted radially until the stylus engages between the thread flanks and the pointer K is opposite to the line mark. ‡ When the pointer is accurately in position, the micrometer reading is noted. ‡ The stylus is then moved along into the next thread space, by rotation of micrometer and a second reading is taken. ‡ The difference between the two readings gives the pitch of the thread. Toolmaker¶s microscope Eye piece Optical head Column ork table ith carriage Hollo base Lamp Collimator lens Mirror Base Toolmaker¶s microscope ‡ Toolmaker¶s microscope is based on the principle of optics. It consists of a heavy hollow base accommodating the illuminating unit. ‡ On the top surface of the base, the work table carriage is supported and its movement is controlled by micrometer screws. ‡ The column carries the microscope unit & various interchangeable eye pieces. ‡ Light from the lamp is collimated and reflected as parallel beam by the mirror. ‡ On its way up this beam collects the image of the object to be inspected and this enters the microscope¶s eyepiece. ‡ A shadow image of the part passes through the objective of the optical head and is projected to a glass screen. Toolmaker¶s microscope ‡ Cross lines are engraved on the ground glass screen which can be rotated through 360o and measurements are made by these cross lines. ‡ Different types of graduated screens and engraved screens are used for measuring different elements. For ex, a revolving screen for measurement of screw threads will contain all the basic profiles standard threads in various pitch ranges & included angles. ‡ Screw thread parameters such as pitch, flank angle, depth of thread,etc. may be measured by matching the projected image of the thread with the master profile obtained from a standard thread. TOOLMAKER¶S MICROSCOPE TOOLMAKER¶S MICROSCOPE Gear tooth measurement ‡ Gears are mainly used for transmission of motion & power and must be of accurate profile to obtain exact velocity ratio. ‡ Two commonly used profiles of gear teeth are the Involute profile & the Cycloidal profile ‡ Involute is defined as the path described by a point on an inextensible cord which is unwound from a stationary cylinder. ‡ Cycloid is defined as the curve traced by a point on the rim of a circle which rolls without slipping on a fixed straight line. GEAR TOOTH NOMENCLATURE GEAR TOOTH NOMENCLATURE ‡ Base circle: It is the circle from which gear teeth profiles are generated. ‡ Pitch circle: It is an imaginary circle which by pure rolling action, would produce the same motion as the toothed wheel. The size of the gear is usually specified by the pitch circle diameter. ‡ Pitch point: It is a common point of contact between two pitch circles of two meshing gear wheels. ‡ Pressure angle: It is the angle between the common normal to two gear teeth at the point of contact and the common tangent at the pitch point. GEAR TOOTH NOMENCLATURE ‡ Addendum: It is the radial distance from the pitch circle to the tip of the tooth. ‡ Dedendum: It is the radial distance from the pitch circle to the root of the tooth. ‡ Face : It is the part of the tooth surface which is above the pitch surface. ‡ Flank : It is the part of the tooth surface which is below the pitch surface. ‡ Circular pitch : It is the distance measured on the circumference of the pitch circle from a point on one tooth to the corresponding point on the adjacent tooth. ‡ Module: It is the ratio of the pitch circle diameter in millimeters to the number of teeth. ‡ Face width: It is the width of the gear tooth measured parallel to its axis. Checking of composite error (Rolling gear test) Gear under test Mechanical dial indicator Master gear heel Fixed carriage Movable carriage Calibrated spring PARKINSON EAR TESTER Rolling gear Test ‡ This test is commonly used in mass production of gear wheels as it takes less time & gives accurate results. ‡ The composite errors can be checked by measuring the variations of center distance when the gear to be tested is rotated under spring pressure against a master gear. ‡ This test reveals any errors in the tooth form, pitch and concentricity of the pitch circle as these errors will cause variation of center distance. ‡ Two carriages are one fixed and the other movable are mounted on the base. The movable carriage is spring loaded towards the fixed carriage. Parkinson gear tester ‡ Two spindles are mounted in a parallel plane on each carriage and are made to suit the bore of the gear wheels. ‡ A dial gauge is made to rest against the movable carriage. ‡ The master gear is mounted on the fixed carriage spindle while the gear to be tested is mounted on the movable carriage. ‡ The dial gauge is then adjusted to zero & the two gears in mesh are rotated by hand and the variations in the dial gauge readings are observed. GEAR TESTER GEAR TOOTH VERNIER CALIPER GEAR TOOTH VERNIER CALIPER ‡ The gear tooth thickness can be conveniently measured by a gear tooth vernier. ‡ Since the gear tooth thickness varies from the tip to the base circle of the tooth, the instrument must be capable of measuring the tooth thickness at a specified position on the tooth. ‡ The caliper has two vernier scales, the vertical vernier is used to set the depth (d) along the pitch circle from the top surface of the tooth at which width (w) has to be measured. ‡ The horizontal vernier scale is used to measure the width (w) of the teeth. GEAR TOOTH VERNIER CALIPER C d 6 W 4 A 3 E D U 0 1 2 4 5 6 2 Pitch circle OA=R O Pitch circle GEAR TOOTH VERNIER CALIPER ‡ Considering one tooth, the theoretical values of w & d can be found which may be verified by the instrument. ‡ The fig shows the chord ADB which is width w but tooth thickness is arc distance AEB. ‡ Also the depth d adjusted on the instrument is slightly greater than the addendum CE, & hence the width w is called chordal thickness & d is called chordal addendum. 360 From the fig, w ! AB ! 2AD. Also U ! where N is the number of teeth 4N 360 In the triangle ADO, w ! 2AD ! 2 v AOsinU ! 2Rsin 4N where R ! pitch circle radius. Pitch circle dia 2R Nvm Module m ! !  R! No of teeth N 2 ¨ 90 ¸ @ w ! N v m sin © ¹. Also from the fig d ! (OC - OD) ªNº Nm m Addendum is equal to one module,   OC ! OE  CE ! (R  m) ! 2 Nm Nm Nm ¨ 90 ¸ ¨ 90 ¸ m Also OD ! RcosU ! cos© ¹   d ! cos© ¹ 2 2 2 ªNº ªNº Nm « 2 ¨ 90 ¸» d! ¬1  N  cos© N ¹¼ 2 ­ ª º½