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Dynamic loads on spur gear teeth. Yeh, Benjamin Hsi-Lu 1959

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DYNAMIC LOADS ON SPUR GEAR TEETH by BENJAMIN HSI-LU YEH B.Sc, Taiwan University, China, 1954 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department of MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1959 ABSTRACT The load on gear teeth while i n operation has long been an open question. So many v a r i a b l e and uncertain factors are involved that i t i s not s u r p r i s i n g that many d i f f e r e n t formulas and rules have been proposed from time to time. Few, i f any, systematic experiments were made to obtain data on t h i s subject u n t i l about 1923, when the ASME Special Research Committee on the Strength of Gear Teeth was organized and the Lewis gear-t e s t i n g machine was b u i l t . The r e s u l t s of t h i s Committee's work were published i n an ASME Research P u b l i c a t i o n , 1931, e n t i t l e d "Dynamic Loads on Gear Teeth". For the experiment i n t h i s t h e s i s , a gear t e s t i n g machine was designed and b u i l t i n order to investigate the dynamic loads on gear teeth. S t r a i n gages were applied to one of the test gear teeth and the main shaft, and were c a l i b r a t e d to give an i n d i c a t i o n of tooth load and shaft torque re s p e c t i v e l y . A s t a t i c tooth load c a l i b r a t i o n on a s t r a i n i n d i c a t o r was made by means of a load arm, a cable and dead weights. The s t a t i c c a l i b r a t i o n helped to set the i n d i c a t o r so that the reading shown on i t during the dynamic load test could be converted to the true load applied. During the dynamic load t e s t of each test s e r i e s , photographs of the o s c i l l o g r a p h traces were taken at d i f f e r e n t machine speeds and at several d i f f e r e n t values of nominal torque. The p r i n c i p a l object of these t e s t s was to determine loads between gear teeth under operating conditions by using s t r a i n gages and e l e c t r o n i c recordings At the present time, the experimental work i s l i m i t e d to involute spur gears only. - i i i -It i s suggested by most gear experts, that the dynamic tooth load may be considered as the sum of the static load and the increment load. The experiments show clearly that the precise gears have no increment loads on their teeth, and that the measured maximum tooth loads are almost the same at different test speeds. The formula for evaluating the dynamic load on spur gear teeth, developed by Earle Buckingham, gives slightly higher values than the measured loads at low speed (approximate pitch line velocity 90 fpm), but diverges widely from the present experimental results at high speed (approximate pitch line velocity 2800 fpm). I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Department Date CONTENTS Acknow1edgement Abstract Chapter I. Brief History II. Description of the Gear Testing Machine III. Calibration and Test Procedure A. Calibration of Torque Measuring Bridge on Main Shaft B. Calibration of Bridge Measuring Tooth Load C. Test Procedure When Measuring Dynamic Load. IV. Experimental Results A. Fir s t Test Series - NRC Involute Spur Gears B. Second Test Series - Canadian Sumner Form Cut Gears C. Third Test Series - Second Pair of NRC Involute Spur Gears. V. Comparison of Calculated Dynamic Loads and Actual Measured Results VI. Conclusions Appendix Bibliography ACKNOWLEDGEMENTS The write r wishes to express h i s gratitude to the s t a f f of the Department of Mechanical Engineering, University of B r i t i s h Columbia, e s p e c i a l l y to Professor W.O. Richmond, for the guidance and help received throughout t h i s research p r o j e c t . This research was made possible by funds supplied by the National Research Council under a grant-in-aid to Professor W.O. Richmond. 0 -1-CHAFTSR I  BRIEF HISTORY1 The dynamic load is the maximum momentary load imposed on the gear teeth by the conditions of service, including the influence of errors in the gear teeth themselves. The nature and extent of these dynamic loads on gear teeth have been studied by many investigators and i t seems that i t s t i l l remains an open question. One of the earliest efforts to account for the increase in dynamic load due to velocity was the use of a velocity factor. Several gears were tested to destruction under static load, and the results compared those from other gears of the same size, strength and material tested to destruction at various speeds. The ratio of the dynamic failure load to the static failure load is called the velocity factor. Oscar Lasche^ first:considered the effects of tooth errors in 1899 and came to the conclusion that the tooth e l a s t i c i t y affected the permissible error. In other words, a greater error is permissible with elastic teeth than with r i g i d teeth, because they can absorb the instantaneous forces without disturbing the rotating masses. In 1908, Ralph E. Flanders discussed the nature of dynamic loads and stated "After some reflection, the writer has come to the conclusion that a variation in the strength of perfectly formed gearing, due to a variation in the velocity, can be due to but one t h i n g — impact caused by the imperfect meshing of otherwise perfectly shaped teeth, deformed by the load they are transmitting". Marx^ ran a number of tests at various pitch line velocities at Stanford University between 1911 and 1915. The results of these tests, (1) emphasized - 2 -the importance of contact ratio (2) questioned the commonly used velocity factors, and (3) indicated that the cutting accuracy had a pronounced effect upon the strength at high speed. Between the years 1900 and 1931 the idea that the dynamic load could be considered as the sum of the transmitted load and ah incremental load "V " . . . . . . . gradually gained support. This equation may be expressed as wd = W + w i where is the incremental load and is due to tooth-form and spacing errors, unbalance, fluctuating loads and the deformation of the teeth under load. W is the transmitted load which is the useful corapoment of force which is transferred from one gear to another during action. Research under the auspices of the ASME Special Research Committee on the Strength of Gear Teeth was organized in 1923. The research was conducted at the Massachusetts Institute of Technology by Earle Buckingham^. Buckingham's report gave the following equation for dynamic loads. W d + y F a < 2 F 2 - F a) where E a is the acceleration load which is practically independent of the applied load and F 2 is the force required to deform the teeth the amount of the effective tooth error. This Buckingham equation generally gives values of the dynamic load which are several times the nominal load. This led Professor W. 0. Richmond to doubt the validity of Buckingham's method and to decide on an experimental program for the purpose of measuring actual tooth loads. Buckingham wrote a mathematical analysis of the mechanics of gearing which was published in his book "Analytical Mechanics of Gears"1- in 1949. Darle W. Dudley commented in his book "Modern Gear Design" 6 — "The Buckingham method gives values of the dynamic load which are slightly high but i t -, i s - 3 -the best method available and w i l l show clearly the effect of masses, shaft stiffnesses, and tooth errors in producing dynamic overloads". Professor w, A. Tuplin ^ has derived a formula for computing the dynamic increment of a pair of high-speed gears from known design factors. His works are based on the wedge analogy for the insertion or withdrawal of error between meshing gear teeth. The equation for dynamic increment force , F, is F = k e„ where k is the stiffness of a two-gear system and e e is the effective error as determined from the design curve, which is used to find the ratio of effective error to actual error from the ratio of time consumed by introduc-tion of error to natural period of vibration. In 1955, Reswick1^ developed an expression for dynamic tooth loads based on a study of a simple physical mechanism which was used to simulate gears in action and gave physical insight into the dynamic behaviour of gears. Reswick*s prediction for dynamic tooth loads are in general, in some cases, agreement with those predicted by Buckingham*^ equation. Professor G.. Niemann and H. R e t t i g 1 1 have made an attempt to measure the actual dynamic tooth forces on a pair of loaded gear with the help of some indicators (which are similar to the function of strain gages). In their experiments the following conclusions were made and the report was published in January 1957« (1) Dynamic incremental tooth forces occur even in the accurate cut gears* (2) with .a small applied load the influence of the dynamic incremental forces is significantly large, (3) increasing the applied load by, for example, 1007. the tooth root stress and tangential tooth pres-sure increase less than 1007., since the dynamic incremental forces do npt increase to the same extent, (4) by increasing the speed, the tooth root stress increases less than the tangential tooth pressure. - A -In order to measure dynamic loads on gear teeth, a dynamical gear testing machine, and a strain gage apparatus, which are very similar in principal with the one devised in this research, were devised by M. UtagawaJ-' in 1956. The testing machine is a power-feed bacK type and has two pairs of gears loaded against each other on two parallel shafts. One pair is test gears and the other is speed up gears. The applied toque is measured by strain gages attached to the torsion shaft. And the strain gage for measur-ing tooth stress is attached to the tooth flank near the f i l l e t curve at the back face of a gear tooth. The stress-cycles or strain-cycles of the tooth are observed through a cathode-ray oscilloscope. In the experiments, Utagawa found that the measured stresses at the spot where a strain gage is attached were nearly equal to the calculated results. Loads and strains are proportional at the same acting point. The frequency of the dynamic load variation almost coincided with the natural frequency of the vibration system formed by the gear and pinion and the mating teeth. Increment load becomes larger as the velocity increases, and the contact ratio has influence on the dynamic load. r - 5 -CHAPTER II  DESCRIPTION OF THE GEAR TESTING MACHINE In order to obtain data on the dynamic loads acting on gear teeth under operating conditions/ a testing machine was constructed in 1956 under the guidance of Professor W, Q. Richmond, Department of Mechanical Engineering, The University of British Columbia. The general arrangement of the testing machine is shown.in Fig. 1. In the machine, the pair of test gears, 'E' and »F», is loaded against a pair of master gears »D». The gears have 24 teeth of 4 diametral pitch. The master gears.are 2 inches in face width and have a 20-degree pressure angle. The machine is designed for test gears of 1 inch face width. 'A.C MOTOR res T GEAR B £XT£NS/ON I SHAFT y TEST 0 EAR> GEAR HOUSING I FLOAT I NO GEAR HOUSING J F>3-1 A SKETCH OF GEAR T65TIN6 MACH/N6 - 6 -The pair of master gears is enclosed in a floating housing 'J* which can pivot about the main shaft 'A' as shown in Fig. 2. Rotation of the housing is prevented by a set screw which contacts a load ring located beneath. Tightening of the screw which bears on the load measuring ring t i l t s the housing thus producing torque in the shaft and tangential load on the teeth of the master gears *D'. Equal and opposite torques are produced in the shafts »A1 and »B». The main shaft 'A' with the test gear on one end and the master gear on the other end is supported by b a l l bearings and is connected to a driving motor by a small coupling 'G*. The machine is driven by a 3 hp induction motor operating at 1780 rpm for high speed tests. For low speed tests, a small \ hp, 1725 rpm, driving through a gear reducer with a reduction ratio of 30 to 1 is used, so that the test gears revolve at about 60 rpm. The driving motor supplies f r i c t i o n and bearing losses. Two secondary shafts, *B' and 'C1, parallel to the main shaft 'A* are connected by an American D - l l coupling *H» of the Oldham type. The larger secondary shaft 'B1 carries the test gear *E» which i s loaded against the other test gear 'F» carried by the main shaft in the gear housing »I 1. At the end of this shaft an attached extension shaft with s l i p rings on i t projects through the plexiglass cover plate. The conductors which transmit the signal of the gear tooth strain gages pass through the core of the hollowed extension shaft and connect to the sli p rings. The smaller secondary shaft •C carries a master gear in the pivotted housing 'J*. The torque in the shaft i s measured by means of four SR-4 strain gages cemented to the shaft. The gages are of type C - l , with a resistance of 502 + 2 ohms, the gage factor is 3.35 + 1%. The gages are cemented at a 45° angle with respect to the shaft axis to measure the principal tensile and compression 6.COO" U .. — m T7^ -Master gear D ^ Mo/n shaft A 'Seccnjor shaft C run ^ ~ **t "* T ^ . V Oil drain plodding gear 1 OH level Section on F/ootmy Gear Housing 'J ' L oad tin a 1 v Fiame of Gear - 8 -strains induced by the torque. The gages .are connected in a Wheatstone-bridge c i r c u i t , as shown in Fig, 3 to s l i p rings which are mounted on the main shaft. Conducting brushes attached to the stationary frame complete the electrical connections between the gages and the external electronic instrumentation. With torsional moments as indicated, gages R-^  and R^  are strained in tension and gages R 2 and R^ in compression. This conventional arrangement of strain gages gives four active bridge legs on the rotating shaft. For measuring gear tooth loads, four SR-4 strain gages of type C-7, with a resistance of 505 + 3 ohms, gage factor 3.27 + 27. are placed at the f i l l e t s of the middle tooth as shown in Fig. 4. The sensitive wires inside the gages are parallel with the line drawn through the corner of the tooth and tangent to the f i l l e t , and the center line of the gage is at the f i l l e t . These gages are connected in a Wheatstone-bridge c i r c u i t as shown in Fig. 4; the leads being brought out to the electronic instruments via the s l i p ring assembly shown in Fig. 5a. The two strain gage bridges are connected through a balancing and calibrating switch to an E l l i s BAl^ L-l strain indicator. The output from the indicator i is displayed on a cathode ray oscilloscope. The switch enables either the shaft torque or the gear tooth load to be selected for display on the screen of the cathode ray tube. The sweep of the oscilloscope is in i t i a t e d , or triggered, by a contact point on the s l i p ring disc for the tooth strain gage bridge (Fig. 5a). The location of this contact point is adjusted so that the sweep starts slightly before the tooth with strain gages makes contact with another tooth. The speed of the sweep is adjusted so that the tooth load record occupies a major portion of the screen. -9 -45°K ft ROT A T/NO f;$. 3 1 — ( f - i SLIP R/HQ o - BAM-I INDICATOR QSC/LLO-GRAPH BRUSH STAT/ONARY F. FRONT OF THE TOOTH B: SACK OF TH£ TOOTH FOR TOOTH <5AQ6S 3- 4 - 10 -To relate the speed of the cathode ray trace to the pitch line velocity of the gears, the device shown in Fig. 5b is used. A voltage pulse is generated in the c o i l when a tooth passes the magnitized core and this voltage variation is recorded to determine the tooth space interval. - 11-^5lip r">$ ^ Fig. 5c? S//p Ring and Carbon Brushes Timing device Wire lead to 06 c /i/o graph ^ \,r> Magnet . P/exi(floss cover of guar housing I | u - • i Test gear £ shaft A •\ j ^ 1 " V Fig. 5b Tooth Speed Measuring Device - 12 -CHAPTER III  CALIBRATION AND TEST PROCEDURE A. Calibration of Torque Measuring Bridge on the Main Shaft. The strain-gage bridge on the main shaft for measuring the torque transmitted by the test gears was calibrated by applying weights at the end of a 20 inch lever. The arrangement of lever and weights is shown in Fig. 6. It was hoped that the torque could also be obtained by measuring the reaction of the pivotted housing containing the master gears by a steel ring with strain gages attached. However, the reading was adversely affected by the adjacent flexible coupling, so that reliable results were not obtained. B. Calibration of Bridge Measuring Tooth Load. The output of the strain gage bridge attached to the root of the gear tooth was calibrated by the arrangement shown in Fig. 6. The output of this bridge is proportional to the bending stress or bending moment at the root of the tooth. This is not directly proportional to load because of the changing moment arm as the contact point travels down the face of the tooth. Thus the calibration had to take into account the position of the force on the tooth as well as the load magnitude. Hence the tooth load conversion charts (Fig. 7) are needed to obtain the load acting on the tooth. For the purposes of reference, gear teeth with strain gages attached may be labelled as shown in Fig. 6, namely, teeth 2A, 1A, 0, 1R, and 2R; where A means teeth ahead of middle tooth 0, and R means teeth Coble r*— Load arm 'a/leu 3 ear ho vs./rig I Line of action fangt to base circle-^ Main Shaft A 6.000 77 Base c/rc &y /Strain cjtfjt S / On Tooth O Test gear E r Drill roa 1 £ i r ig Secondary \ Shaft B ' ' Centre heaht obeve the reference <surfoce Ca/cuIo7g this distance af>d moke bloc A to suit Load oppiied at pitch /,/ line on Tooth IA / Stock (grind'c/eoraoce as necessary f 1—, _ L Finished surfaces Static Ca/ibrat/on of QridjQ M e a ' a r / n ^ Tooth Load r i 9 - 6 - 14 -behind the middle tooth. Strain gages on the tooth 0 were connected in a bridge ci r c u i t (see Fig. 3). This connection gives almost no reading for the loads applied to teeth 2A, 1A, 1R, and 2R, but a maximum reading for the loads applied to tooth 0. Loads tangent to the base c i r c l e of the gear and normal to the tooth surface were applied at the tip of the tooth, on the pitch line, and at the point on the tooth profile at the end of the tooth engagement. The loads were applied by putting a special designed steel block together with an eighth inch diameter steel d r i l l rod underneath the tooth as shown in Fig. 6. The total height of the block and the rod was accurately calculated for each of the three positions previously described. During static calibration adjustment had to be made in the coupling to allow the gear to take up the desired position and yet maintain the loading arm at right angles to the cable supporting the weights. The tooth load conversion charts for different pairs of gears were plotted on a tooth contact cycle basis. The vertical scale of the oscillograph display w i l l have to be corrected by this chart to give the true tooth loads. A sample of the tooth load conversion chart i s shown in Fig. 7. C. Test Procedure When Measuring Dynamic Load. The following procedure was followed in the dynamic load tests. After the machine was started the desired load was applied u t i l i z i n g shaft torque measurement as a guide. The readings for each run included the speed, the average torque and oscillograph records of the shaft torque variation, the dynamic load record from the tooth, and a record - 15 -indicating the tooth speed. Tests were run at two speeds, slow operation at about 60 rpm (92.4 fpm pitch line velocity) and fast operation at nearly 1800 rpm (2825 fpm pitch line velocity). Records were taken for torque values of 500, 1000, 1500, and 2000 in-lbs. Tests were carried out on three pairs of gears, two pairs were 20° f u l l depth involute cut by the National Research Council, and one pair was 14%° f u l l depth cut by the form cutter method at the Canadian Sumner Iron Works. One pair of NRC gears had bushings installed in the bores, which made the tooth pitch c i r c l e eccentric to the bore of the gears. A/ore.. Loads measured normo/ TO TOOTh iSUr-foce. pCIHl OF CONTACT 2 pAlfiS OF TEETH CAKF-VIN* THt LOAD CONTACT CYCL& Fig 7 TOOTH LOAD CONVERSION CHART OF TH£ F/RS7 PAIR of NRC MAD£ 20-DEG. INVOLUTE SPUR G£ARs Point OF-CONTACT 2 PAIRS OF TEETH CAMY"*, THE LOAD Point TtMTH CARHY/H6 - 16 -CHAPTER IV  EXPERIMENTAL RESULTS First Test Series - NRC Involute Spur Gears (20-degree pressure angle) This test series was run in two parts, f i r s t with the gears set so that the eccentricity of the bores were in opposite directions, and second with the eccentricity in the same direction. a) Machine at low speed The typical records for the f i r s t test series are shown in Fig. 8. Fig. 8a shoxas a typical shaft torque variation record obtained from the strain gages on the main shaft. The straight line at bottom of the diagram represents the zero reading of the instrument. The upper curve is the torque variation under a measured torque of 1950 in-lb., when the gears were rotating at 56 rpm (88 fpm pitch line velocity). The nature of the torque variation for a f u l l revolution is shown in Fig. 8b. The tooth load variation obtained at 88 fpm pitch line velocity under a measured torque of 1950 in-lb. i s shown in Fig. 8c. By using the tooth load conversion chart (Fig. 7) the corresponding acting tooth load diagram was plotted in Fig. 8d. In the diagram, i t was found that the load on tooth 0 where the strain gages were attached increased gradually from zero value at the i n i t i a l point of contact to a maximum value which was achieved when tooth 0 carried the f u l l load. The load decreased gradually as the pair of following teeth came into contact, and the load became zero at the fi n a l point of contact. The similar tooth load diagrams (Fig. 10a and 10b) were obtained for the second test series at a - 17 -measured torque of 2020 in-lb. As the torque values were reduced, the shaft torque variation maintained the same pattern as shown in Fig. 8a and 8b. However, the tooth load curve shows an increase in the duration of maximum load as illustrated in Figs. 8c to 8j. At higher load, the maximum tooth force only appeared during the period when one tooth carried the whole load* As the torque was reduced to a lower value, the tooth load reached a maximum value earlier and this condition persisted for nearly half the contact cycle. Machine at high speed. A l l the records obtained at high speed (Fig. 9 and Fig. 11) have the same pattern as at low speed, but vibration effects were present. The upper curve in Fig. 9a shows a typical record of shaft torque variation obtained at 1775 rpm (2786 fpm pitch line velocity) under a measured torque of 1600 in-lb. The lower curve in the same diagram represents the zero reading of the instrument. The shaft torque pattern of second test series at a measured torque of 2320 in-lb. i s shown in Fig. 11a. The irregularity of the zero curve may be explained as follows: F i r s t l y , the s l i p rings on the main shaft are composed :of two halves with the result that a gap and an irregularity exists at the junction of the two halves. The discontinuity of the rings produces vibration of the brushes and a variation of the brush pressure. At 60 rpm this change in pressure is very small. Since the change of pressure i s proportional to the square of the speed, - I S -the change in pressure at 1800 rpm is 900 times as great as at 60 rpm. This pressure vibration causes the contact resistance to vary. This is one of the cause of ripples in the zero curve. Also the ripple is much effected by dust on s l i p rings — clean rings give a smoother signal. Secondly, generator effect may have had some influence. The four strain gages which compose the Wheatstone-bridge c i r c u i t , form a closed c i r c u i t . If the closed ci r c u i t cuts a magnetic f i e l d (the earth's f i e l d or a f i e l d due to the current in nearby conductors) then voltage w i l l be set up between the terminals of the c i r c u i t . The magnitude of this voltage was not determined. Fig. 9b shows the shaft torque variation for a f u l l revolution under a measured torque of 1600 in-lb. The tooth load variation at high speeds is illustrated in Figs. 9c to 9h. Figs. 9c and 9d represent the tooth load variation under a measured torque of 2200 in-lb. Reducing the nominal torque value to 1000 in-lb., more ripples appeared in the tooth load pattern and a longer period of maximum load is evident as shown in- Fig. 9h. The ripple represents the natural frequency of elastic system which is 2770 cps by measuring from the record. The typical records of second test series at high speed are shown in Figs. 11a, l i b , and 11c. For these records the machine was run at 1795 rpm (2820 fpm pitch line velocity) with a measured torque of 2320 in-lb. As .mentioned earlier, the oscillograph records were analysed by plotting maximum measured tooth loads against the torque applied - 19 -on the shaft to the same time scale, as shown in Fig. 12. With the eccentricity of the gears in opposite direction curve 2 was obtained. Curve 2 represents the test series run at low speed, and curve 3 i s the high speed result. Both curves are f a i r l y close to the theoretical prediction (curve 1) in which the forces on the tooth are equal to the torque applied divided by the base c i r c l e radius of the gear. For the same torque value, the maximum tooth loads at low speed are higher than the theoretical curve while the maximum tooth loads at high speed are slightly lower than the theoretical curve. With the eccentricity of the gears in the same direction curve 5 for low speed, and curve 4 for high speed were obtained. The maximum tooth load at low speed (curve 5) was slightly greater than at high speed (curve 4) the same as in the previous test. Both curves '4 and 5 are farther from the theoretical curve and this may be due to interference between teeth. - 20 -•/ Tooth Space 2 . 2<90O //>-/© 0 (.-typical) Hm 9I8 Fig.8a Shaft torque variation under a measured torque of 1950 in-lb at low speed. (Test No. 5) M M I M l s — 1 SSSSS5SSSS 2 — " * 'Fig. 8b Shaft torque variation for f u l l revolution at 1950 in-lb at low speed. (Test No. 5) - 21 -• / Tooth Space 1000 lbs— 740 lb {typical) Pig.8c Tooth load variation at a measure torque of 1950 in-lb at low speed. (Test No. 5) Figo8e Tooth load variation at a measured torque of 1500 in-lb at low speed (Test No. 5) Fig.8g Tooth load variation at a measured torque of <U00 in-lb at low speed. (Test No. 5) — CONTACT CYCLE. • (l.Sf TOOTH SfAZS) Fig.8d Force diagram of Pig. 8o. 5-75 lb F i g . 8f Force diagram of Pig 08e. Fig.8h Force diagram of Pig.8g. Fig. 81 Tooth load variation at a measured torque of 410 in-lb at low speed. (Test No. 5o) (Double Scale) -/. 34 TOOTH SPACe Pig. 8j Force diagram of Fig. 8i - 22 -Fig. 9a Shaft torque variation at a measured torque of 1600 in-lb at high speed (Test No* 6) Fig. 9b Shaft torque variation for f u l l revolution at 1600 in - l b at high speed. (Test No. 6) - 23 --/ •footh .space, -fyp ISo it, looo lb — 4bo it 0 •fyp ICO I Fig.9c Tooth load pattern at a measured torque of 2200 in-lb at high speed. (Test No. 6) CCNlACT PT- | CONTACT CYCLE {1-6 t TOOTH SPACE) Pig. 9d Force diagram of Fig. 9c. - 24 -JOOO /bi 0 CCNTAZ.T CYCLE (7.SS 7C0TM SPA-E) Pig. 10a Tooth load p a r t e m at a measured F i g . 10b Foice toriue of 2020 i n - l b at low .:peed. diagram of (Test No. S) F i g . 10a o 2000 >'n-lb /OOO O F i g . 10e Shaft torque pattern at a measured tarquo of 2020 in-lb at low speed, (fest No.8) / - 25 -y^T^ tooth spoce, -fyp. /OOO lbs I t Q: - J ;> ii o *5< T / J y PITCH PClNT CONTACT CYCLE (I.S7 TOOTH SPACE) Pig.11a Tooth load pattern at a measured torque of 2320 in-lb at high speed (Test No. 7) Pig. l i b Force diagram of Fig. 11a. 2000 i*-lb /OOO Fig. l i t Shaft torque pattern at a measured torque of 2320 in-lb at high speed (Test No. 7) - 26 -1000 vo -J Q § S o o K I 3 § 5 800 600 400 200 MEASURED TORQUE VARIATION ON SHAFT (in-lbs) (Torque values areon the same time scale as tooth load) Fig. 12 Analysis of oscillograph records with the f i r s t pair of NRC gears (20-deg. pressure angle) Curve 1 Theorteical tooth load curve Curve 2 Maximum eccentricity vs minimum eccentricity at low speed. Curve 3 Maximum eccentricity vs minimum eccentricity at high speed 0 Curve 4 Both maximum eccentricities meshed at high speed. Curve 5 Both maximum eccentricities meshed at low speed0 - 27 -Second Test Series - Canadian Sumner Form Cut Gears (14% degree  pressure angle) The second test series were conducted according to the same general procedure as detailed for the f i r s t test series, with the exception that after each test, one of the gears was rotated one third of a turn so that a different tooth on one gear was meshed with the tooth on the other gear to which the strain gages had been attached. This was done in order to compare the characteristic of the load pattern for different pairs of mating teeth. The results of the three series of tests are shown in figures 13 to 18 for the different meshing conditions. From the characteristic loading patterns, one can see that one tooth only carries the load, there is no sharing of load as in more accurately cut gears. The vibration is pronounced in the high speed tests. Essentially similar load patterns were obtained for a l l three meshing conditions. The records were analysed by plotting curves with the torque variation on the shaft as the abscissa, and the maximum true tooth loads as ordinates. In Fig. 19, curve 1 represents the theoretical tooth load which equals the applied torque divided by the base c i r c l e radius of the gear. In running the test, the pair of teeth in contact were marked ?A* and curve 2 was obtained at low speed, then one gear was shifted by one third of a revolution so that the tooth marked 'B' on the driving gear was in contact with tooth 'A* on the driven gear, and curve 3 was obtained at low speed. Similarly, curve 4 was obtained when a pair of teeth marked 'C* and 'A1 were meshed together. - 28 -Curves 2, 3, and 4 which are plotted from the experimental data, diverge appreciably from the theoretical prediction (curve 1). This is very likely due to calibration errors. At-high speed, severe vibration is evident in the records thus making analysis d i f f i c u l t . From a comparison of measured maximum tooth loads in the experimental results, i t can be seen that the Canadian Sumner Gears have far greater tooth loads than the NRC gears. When analysing the tooth load records as a f i r s t approximation, the tooth can be considered to be an undamped linear single-degree of freedom system subjected to a. load which is gradually applied and then maintained indefinitely. The detailed analyses are given in Chapter V. \mwri_ 111 II m m Fig. 14« Shaft torque variation at a •MMMMd torque of 1620 in-lb a t high speed (Teat Ho. 10) / t o o t h s p a c e j ~fyp. - 30 -1 3 11 II •1 S 3 11 fe sas Fig. 13b Tooth load pattern at a measured torque of - 2000 in-lb at low speed. (Test No. 9) OOO lb 850 corrrACT cycle \ (leg -roerk SPACE)' Fig.13c Force diagram of Fig. 13b Fig. 13d Tooth load pattern at a measured torque of 1450 in-lb at low speed (Test No. 9) ••ii m g • • • l l S I M S S S ! Fig. 13f Tooth load pattern at a measured torque of 1050 in-lb at I o n speed (double scale) 62o lb Fig. 13e Force diagram of Fig. 13d lb i WlM TOOTH SPACED i&o Fig. 13g Force diagram of Fig. 13f Fig. 13h Tooth load pattern at a measured torque of 560 in-lb at low speed (double scale) JU- f.o4 TOOTH S W £ - * | Fig. 13i Force diagram of Fig. 13h fooo lbs - 31 -/ tooth space, typ. ar I l l f l V I I IfllTLlHMHI n u n • • — A Fig.14b Tooth load pattern at a measured torque of 1620 in-lb at high speed (Test No. ic) • • • [ ! • • • • • • • • • • l i m n H I PI • n i i i i i • 1IFIIIUI II ri<i.vi • i i i i i i ••••••••luw'ap bssSS Fig. 14d Tooth load pattern at a measured torque of 1500 in-lb at high speed (Test No. 10) 7c e Contact cYcie Fig. 14c force diagram of Fig. 14b. (frequency = 2860 cps) IZio>b io tooth cya.e-Fig. 14e Force diagram of Fig. 14d. m i •iniir"! ll'iriua* Fig.l4f Tooth load pattern at a measured torque of 1000 in-lb at high speed (Test No. 10) >iio lb -/.»7 Teem SPA Fig. 14g Force diagram of Fig. 14f. da The tooth narked A on the driven gear meshed with the tooth B on the driving gear. The records in this case were of the pattern as in the preceding case, so only some typical diagrams are shown below0 a) Machine at low cpeed. /OOO lbs 0 / tooth space, "fyp. Pig. 15c Shaft torque variation at a measured torque of 2000 in-lb at low speed (Test No0 12) - 33 -b) Machine at high speed,, Some typical records are shown in the following diagrams. The measured frequency of the tooth load pattern i s 3000 cps. / tooth space, typ. /OOO /As-lli I  II m i m i i n iii 11 fir II nun 11 H M i f i m II i w Fig. 16a Tooth load pattern at a measured torque of 2000 in-lb at high speed. (Test No. 11) <0 vi K |! u is I2CC liSO It n 1 /100 vi n 10 CONTACT CYCt£ tU7 -TOOTH SPAClf*" toe Fig. 16b Force diagram of Fig. 16a. 2000 in-lb /OOO mm*mmm i f ZjM ~\1M A W * ™ * 1 Fig. 16c Shaft torque variation at a measured torque of 2000 in-lb at high speed (Test No. 11) - 34 -3. The tooth narked A on the driven gear meshed with the tooth C on the driving gear» Some typical diagrams are shown below. a) Machine at low speed. / too/h space, fyp. JOOO lbs - H 1 1 i 1 • i l i i Pig. 17a Tooth load pattern at a measured torque of 1920 in-lb at low speed. (Test No. 13) Pig. 17b Force diagram of Fig.17a. Fig. 17c iihaft torque variation at a measured torque of 1920 in-lb at low speed. (Test No. 13) 35 -b) Machine at high speed. The measured frequency of the tooth load pattern and the shaft torque pattern were 3000 and 2890 cycles/sec respectively. fOOO lbs 2000 in -lb /OOO i2oo Fig. 18a Tooth load pattern at a measured torque of 2000 in-lb at high speed. (Test Ho. 14) Contact cycls I 1.12 TOOTH SPAi-S Fig. 18b Force diagram of Fig. 18a. / tooth space, typ. 0 ! H B B — I B w a u l — Fig. 18c Shaft torque variation at a measured torque of 2000 in-lb at high speed. (Test No. 14). 36 -0 500 1000 1500 2000 MEASURED TORQUE VARIATION ON SHAFT (in-lbs) (Torque values are on the same time scale ae tooth load) Fig. 19 Analysis of oscillograph records with the Canadian Sumner Gears (14 l/2 deg. pressure angle) Curve 1 Theoretical tooth load curve,. Curve 2 Tooth 'A' meshed with tooth 1A' at low speed,, Curve 3 Tooth 'A' meshed with tooth 'B' at low speed Curve 4 Tooth 'A' mashed with tooth '6' at low speedo - 37 -C* Third Test Series - Second Pair of NRC Involute Spur Gears (20-degree  pressure angle)" The third test series were conducted according to the procedure detailed for the f i r s t test series with the exception that after each test one of the gears was rotated one third of a turn so that a different tooth on one gear was meshed with the tooth on the other gear to which the strain gages had been attached. The results of the three series of tests are shown in figures 20 to 25 for the different meshing conditions. The torque variation obtained in these tests fluctuates less than the f i r s t pairs of NRC gears and closes to a straight line. The reduced torque fluctuation tended to suggest that the second pair of NRC gears had fewer tooth errors than the f i r s t pair. From the tooth load recordings one can see clearly the load i s shared by the teeth during the beginning and end of the contact cycle. At high speed the oscillograph recordings have the same general pattern as at low speed, except that there were some vibration effects. Referring to Fig. 21d, hi represents the average force on a pair of teeth . which are approaching the end of contact. The sudden reduction in force represented by the line gh indicates that a new pair of teeth have come into contact at 'a 1 and have assumed a portion of the load. Results of experiments indicate that a new pair of teeth must suddenly come into contact causing a very rapid increase in force (curve ab, Fig. 21d). The line cd shows that the load was shared by two pairs of teeth until the load jumped to i t s maximum value when one tooth started to carry the whole load. When the action passes point e, the vibrating - 38. — motion appears to decay and settle down to an average maximum value represented by fg. The oscillograph records obtained from the tests of the second pair of NRC gears were analysed and their results are shown in Fig. 26. In the diagram, curve 1 represents the theoretical tooth load which is equal to the applied torque divided by the base c i r c l e radius of the gear. Curve 2 resulted with both 'A* teeth of the pair of gears meshed together at low speed while curve 3 is the result at high speed. Curves 4 and 5 were obtained at low speed and at high speed respectively with one gear shifted one third of a revolution. Similarly, curves 6 and 7 were obtained after one gear was again shifted by one third of a revolution. In these tests the majority of the measured tooth loads were lower than the theoretical load for the same applied torque. This is either due to the calibration errors, or could be explained by the Reswick's1^ study in the following paragraph. Also, the larger the torque value, the greater the range between the theoretical value and the measured load. For the same pair of meshed teeth the measured tooth loads at low speed are sometimes higher than the loads at high speed; whereas in the tests of the f i r s t pair of NRC gears, the tooth loads at low speed always had a slightly higher value than at high speed. J. B. Beswick*^ developed an expression for dynamic tooth loads based on a study of a simple physical mechanism. "From f i r s t principles'^ stated Reswick, "equations have been developed which predict values for dynamic tooth loads that are in general agreement with those predicted by Buckingham*^ equation for 'lightly loaded gears'* but differ somewhat in the case of 'heavily loaded gears'"". Results developed - 39 -from experiments of the model indicated that in many heavily loaded gears the total dynamic load may be actually less than the static load • determined from the transmitted power, and the dynamic loads may be ignored, but the dynamic load is important in lightly loaded gears. + The term "lightly loaded gears" applies to those gears in which the deflection due to the steady power load is very much smaller than the manufacturing error. In this case, only the pair of teeth involved in the engagement is deflected. * The term "heavily loaded gears" is applied to those gears in which the deflection due to the steady power load is equal to or greater than the manufacturing tooth error. Under these conditions the load is carried by two pairs of teeth during tooth engagement. 4 0 -/ tooth spcrce, -fyp. Fig. 20a Shaft torque variation at a measured torque of 1980 in-lb at low speed. (Test N o . 16) H u m u s ! !••••••••••• 181 181 • i i M ! 8 f i f i s a a » J S - S S a l i ' • •—•—• •—— - w —*—• Fig. 20b Shaft torque variation for f u l l revolution at 1980 in-lb at low speed, (Test No. 16). J - 41 -500 O lb / tooth space, typ, BSSHS Fig. 20c Tooth load variation at a measured torque of 1980 in-lb at low speed-double scale (Test No.16) p, rcn pmM -cchtact crcte — # H O&3rev rft SPA^S) Fig .20d Force diagram of Fig. 20Co Fig.20e Tcoth load variation at a measured torque of 1450 in-lb at low speed-double scale(Test No.16) ••• M l ••• •1 mm * J ••• • • i n • mm •• Fig.20g Tooth load variation at a measured torque of 960 in-lb at low speed-quadruple scale.(Test No.16) f i | — t.62 7007H $PA<*£ —-| j Fig. 20f Force diagram of Fig.20e. I.b2 TOOTH 5WCf_ Fig o20h Force diagram of Fig. 20g. Fig. 20i Tooth load variation at a measured torque of 500 in-lb at low speed-quadruple scale.(Test No.16) Fig.20j Force diagram of Fig. 20 i . - 42 -/ tooth space, typ. 2000 j i ^ p p i i i i m i i i i r 1 1 I /OOO (••••••••••••••••MB (••••••••••••••••MM • l l R I I I I P r A Fig. 21a Shaft torque variation at a seasured torque of 2000 in-lb at high speed. (Test Ho. 17) P w n j I I I I H I I i n i P ••••••rj Fig, 21b Shaft torque variation for ful l revolution at 2000 in-lb at high speed. (Test Ho, 17) r - 43 -Soo /6s — / tooth space, fyp. i i i i i r i i i i i IBdVILall M l l i l l l l l l l Fig 021c Tooth load pattern at a measured torque of 2000 in-lb at high speed-double scale.(Test No017) lid mi :ii Pig. 21e Tooth load pattern at a measured torque of 1450 in-lb at high speed-double scale (Test No. 17) ••••II ir INITIAL W T C M > r F'"+L „ P CONTACT cr<:«-E -i d.bi TOOTH SPACS ; 1 Pig. 21d Force diagram of Fig. 21c Fig. 21g Tooth load pattern at a measured torque of 1050 in-lb at high speed-quadruple scale. (Test No.17.) 47c 42.0 J«« r\ [\J i 7 * y 25° ito US j li' _ ,,(,/ TOOTH 5/MCE — Fig. 21f Force diagram of Fig. 21e. Fig. 21h Force diagram of Fig. 21g i i r i i n m i i i i i i i i i i i i t r i i i n i ! Siiiiiiiiiniiit Fig. 2 l t Tooth load pattern at a measured torque of 570 in-lb at high speed-quadruple scale.(Test No.17) lie /5. M t\/\ f\ J 0 /.S9 TOOTH SPAC& M Fig.21j Force diagram of Fig. 2 l i - 44 -2. The tooth marked A on the driven gear meshed with the ( tooth B on the driving gear. a) Machine at low speed / tooth space, fyp. 500 ibs Pig.22a Tooth load variation at a measured torque of 1450 in-lb at low speed-double scale (Test No.19) HTCH VtlHT CONTACT CYCLE_ /•6i TOOTH SP4C£ Fig.22b Force diagram of Fig. 22a 200O in-/b /OOO wm • • r • • • mi PI m m • w m m •1 I • m m m •1 •1 Fig.22c Shaft torque variation at a measured torque of 1450 in-lb at low speed (Test No. 19) fif > 3 P.6J - 45 -b) Machine at high speed. / -tooth spacet typ. •Soo ju5 I Iff.HMI I I I ? l i m I I I 'Until ••I lUHIJI'UI l l l l l l l l l l l l l l l l l l l l l l i m i i i i m i m ^ i n i • Fig. 23a Tooth load pattern at a meas-ured torque of 1550 in-lb at high speed-double scale. (Test No. 18) f\s,c A K 4So\ \ A / \ 1 \ \ J 4oc 4eo \j \ no V A I L ' i}« \ A IJ*\ CCNTACl CYCLE (l.6o TOOTH SPACE) Fig. 23b Force diagram .of Fig. 23a 2000 in-lb IOOO ••••••••• J Fig. 2 3 c Shaft torque variation at a measured torque of 1550 in-lb at high speed. (Test No.18) 3. The tooth marked A on the driven gear meshed with the tooth C on the driving gear. The results of this test were very similar to the f i r s t test series in which both "A" teeth were meshed together. One typical recording for each run is shown in the following diagrams. a) Machine at low speed / tooth space, -fyp. 500 I I I H I I l l l l l i i i i i i m i m L k l l l l !••••! b) Machine at high speed. i i i i i i i i i i i i I^TIUll »•!•• wmrnm m \ IIIL^IIIIIIIII) • • > W | 1.^1111 • • • i WmmM ~uW9*mf M\Z-Z,mmmmmmt ingi 11 i n • «i!l9lll Fig. 25a Tooth load pattern at a measured torque of 2220 in-lb at high speed-double scale (Test No. 21 Fig.24a Tooth load pattern at a measured torque of 1800 in-lb at low speed-double scale. (Test No. 20) CONTACT CYCLE_ C/.42 TOOTH SPACE) Fig.24b Force diagram of Fig. 24a I I if. 1 A e / m 4ie ! i \ il» **° \ /*\ f 2iu i CONTACT CYCLB (I.b4 TOOTH SPAc£~) *" Figo25b Force diagram of Fig.25a. (frequency 5100 cps) - 47 /OOO C D o K o o I Q Uj ;> a Q $00 /OOO ,500 2000 2500 MEASURED TORQUE {/AR/ATION ON SHAFT (IN-LBS) (TORQUE l/ALUES ARE ON THE SAME T/Me SCALE AS TOOTH LOAD ) Fig 26 ANALYSIS OF OSCIULO QRAPH RECORDS-, W/TH THE SECOND PAIR OF NRC OEARS (2.0-DEO. PRESSURE ANGLE) Curve Curve Curve Curve Curve Curve 6 Curve 7 1 2 3 4 5 Theoretical tooth load curve. , Tooth »A» meshed with tooth,'.A'1 Tooth •A1 meshed with tooth 'A' Tooth 'A' meshed with tooth 'B» Tooth 'A' meshed with tooth 'B' Tooth 'A' meshed with tooth 'C« Tooth 'A' meshed with tooth 'C at low s pee do at high speedo at low speedo at high speedo at low speedo at high speedo - 48 -CHAPTER V COMPARISON OF CALCULATED DYNAMIC LOADS AND ACTUAL MEASURED RESULTS Buckingham's Formula Ea r l e Buckingham developed a general formula f o r c a l c u l a t i n g the dynamic load on the spur gear tooth. This formula covers average conditions rather,than s p e c i f i c ones, p a r t i c u l a r l y where the values of the many factors are unknown or indeterminate. Variable Values f o r Test Gears rcip = mass e f f e c t of pinion at R^ nig = mass e f f e c t of gear at R 2 Rl = p i t c h radius of pinion = 3 i n . R 2 = p i t c h radius of gear = 3 i n . Nj_ = number of teeth i n pini o n « 24 ^ 2 = number of teeth i n gear => 24 P = c i r c u l a r p i t c h = — — = P d 4 F = face width of gears = 1 i n . V => p i t c h l i n e v e l o c i t y ft/min (See Table I) C m = mass f a c t o r , which depends on the p i t c h - l i n e v e l o c i t y (See Table I) P = P2N XF 64 .002 K\ + 1 .001 R^ + 1 0.233 - 49 -g 64 .002 R* + l .001 R^ + 1 0.233 Therefore, the e f f e c t i v e mass at the p i t c h l i n e of the gear i s : m m m m _P 8 m P + m g ; 0.1166 C m TABLE I Mass Factors - Values of C m (See Reference 8) Gear Speed c RPM P i t c h - l i n e v e l o c i t y ft/min m 55 86.5 8.448 56 88.0 8.420 57 89.5 8.393 58 91.1 8.364 59 92.7 8.336 60 92.4 8.307 1775 2786 1.327 1780 2796 1.325 1785 2804 1.323 1790 2811 1.321 1795 2820 1.320 1800 2825 1.319 - 56 -Let W = total applied load, lbs. = 33,000 hp ,. , . — v ~ — *- = applied torque pitch c i r c l e radius C = deformation factor, depending upon tooth error and element of gears (See Table II) TABLE II Values of Deformation on Factor C (See Reference 8) Material Tooth Form Error in Action (in) .0005 .001 .002 .003 .004 .005 Steel & Steel 14 % deg. 800 1600 3200 4800 6400 8000 .20° Full Depth 830 1660 3320 4980 6640 8300 Then f j . .0025 R + R K l 2 R1 R2 2 2 . mV .000129 C V m f 2 = FC + W f l f2  f l + f2 The total dynamic load is W d - H + / f a < 2 f 2 - f ) - 51 -1. First pair NRC Spur Gears: Assuming the gears were "precision gears," the tooth error is .0006 inch, therefore from Table II, C » 996. TABLE III Data for Evaluating dynamic loads, a) Machine at low speed (Test No. 5) Meas. Torque in-lb Meas. rpm V ft/min Cm m *1 W lb *2 f a W d lb Meas. max. tooth loads lb 1950 1500 1100 410 56 56 57 58 88 S8 89.5 91.1 8.420 8.420 8.393 8.364 .982 .982 .978 .975 0.44 8.44 8.70 9.00 650 500 367 137 1646 1496 1363 1133 S.40 8.39 8.65 8.92 ei6 or-s 520 280 725 575 410 190 b) Machine at high speed (Test No. 6) 2200 1600 930 1775 1780 1780 2786 2796 2796 1.327 1.325 1.325 .1547 .1545 .1545 1334 1338 1338 733 533 310 1730 1530 1305 753 712 660 2160 1840 1445 750 570 310 Table III shows that the measured maximum tooth loads at high speed are very similar to those at low speed, and in both speeds the measured tooth loads are slightly higher than the static load, w, determined from the transmitted torque for the same applied torque. The measured tooth load in the tests at low speed is nearly 117. greater than the static load. At high speed, the measured tooth loads are very close to the static load. Therefore, in these tests the dynamic loads may actually be neglected. The value of the dynamic loads calculated by Buckingham's formula at low speed showed a 12.5% larger value than the measured load at a torque of 1950 in-lb and were 47% greater at 410 in-lb. At high speed, Wrf, is 190- larger than the measured load at a torque of 2200 in-lb and increased to 3607. at 930 in-lb. - 52 -TABLE IV The experimental results of the f i r s t pair of NRC gears. Maximum eccentricity of one gear i a meshed with the minimum eccentricity of the other. Machine at low speed (Test No. 5) Meas. Mea8.Max0 Length of Torque at Equivalent Torque tooth load contact cycle nominal value Max. tooth in-lb lbs in tooth space load. 1950 725 1.59 2000 744 1500 575 1.53 1500 575 11U0 410 1.46 1000 373 410 190 1.34 500 232 Machine at high speed (Test No. 6) 2200 750 1.61 2000 682 1600 570 1.44 1500 535 930 310 1.60 1000 334 —— — 500 — Maximum eccentricities of both gears meshed together. Machine at low speed (Test No. 8) 2020 790 1.55 2000 782 1520 600 1.55 1500 593 1050 460 1.49 1000 438 550 260 1.45 500 236 Machine at high speed (Test No. 7) 2320 900 1.57 2000 775 1590 610 1.56 1500 575 1100 450 1.57 1000 410 640 300 1.40 500 234 - 53 -2. Second pair of NRC Gears: This pair of NRC gears were "precisely ground gears" with an estimated tooth error 0OOO6 inches, Table II gives C = 996. TABLE V Data for evaluating dynamic loads. a) Machine at low speed (Test No. 16) Meas. Torque in-lb Meas rpm V f t /min. C m m *1 W ' f 2 K W d lb Meas. Max.Tooth Loads l b . 1980 56 88.0 8.420 .982 8.44 660 1656 8.39 82 6* 610 1450 57 89.5 8.393 .978 8.70 m 1480 8.65 M l 450 960 57.5 90.3 8.379 .977 8.85 320 1316 8.80 472T 300 500 58 91.1 8.364 .975 9.00 161 1163 8.94 310, 161 B) Machine at high speed (Test No. 17) 2000 1780 2796 1.325 .1545 1338 667 1663 742 2050 650 1450 1785 2804 1.323 .1543 1342 483 1480 703 1740 470 1050 1790 2811 1.321 .1541 1347 350 1346 673 12H 320 m 1790 2811 1.321 .1541 1347 190 1186 630 1200 150 The test results indicate the interesting fact that not only are dynamic loads non-existent, but also that the measured maximum tooth loads are slightly lower than the static loads. Reswick's 1 0 experiments demonstrated that i n many heavily loaded gears the total dynamic load may be actually less than the static load determined from the transmitted power0 Therefore, i t would appear that there i s existing evidence in support of the present results. The calculated dynamic loads, Wd, by Buckingham's formula, are larger than the measured loads at low speedo The percentage of larger value from a torque of 1980 in-lb to 500 in-lb are 35$, 43$, 57$ and 88$ respectively. For the machine at high speed, the deviations of the experimental results from the Buckingham predictions are even greater. The theoretical values of are greater than the measured loads by. the following percentages 215$, 270$, 375$ and 760$ progressing from the highest to the lowest torque value respectively. - 54 ~ TABLE VI The Experimental Results of the second pair of the NRC Gears. Machine at low speed (Test No. 16) Meas. Meas.max. Length of Torque at Equivalent Torque tooth load contact cycle nominal maxotooth in-lb lbs in tooth space. value load.lbSo 1980 610 1.63 2000 616 1450 450 1.615 1500 465 960 300 1.62 1000 313 -~ 500 165 1.61 500 165 Machine at high speed (Test No. 17) 2000 650 1.63 2000 690 1450 470 1.61 1500 485 1050 320 1.59 1000 305 570 150 1.59 500 132 Low Speed(Test No. 19) 2000 600 1.63 2000 600 , 1450 480 1.62 1500 496 1000 350 1.61 1000 350 510 200 1.60 500 196 < High Speed (Test No. 18) 2000 750 1.64 2Q00 750 1550 600 1.60 1500 580 1070 450 1.60 1000 420 500 275 1.58 500 275 Low Speed (Test No. 20) 1800 520 .1.62 2000 578 1300 390 1.61 1500 450 1000 290 1.62 1000 290 500 140 1.58 500 140 High speed (Test No. 21) 2220 600 1.64 2000 540 1400 320 1.61 1500 • 343 960 250 1.60 1000 260 480 — . — 500 —— - 55 -3. Canadian Sumner Iron Works Ltd. Involute Spur Gears (14 h deg. Pressure Angle) Assuming these gears were "First class commercial gears," the error in action is 0.0026 inch, and from Table II, C <- 4160. TABLE VII Data for Evaluating Dynamic Loads, a) Machine at low speed (Test No. 9) Meas. Meas. V Cm m f l W f w d Meas. Torque rpm ft/min lb 2 a max. tooth in-lb loads lbs. 2000 56 SCO P. 420 .982 C.44 6o7 4?25 8.43 950 1000 1450 57 89.5 8.393 .978 8.70 483 4643 8.69 765 710 1050 57.5 90.3 8.379 .977 8.85 350 4510 .8.84 630 490 560 50 91.1 8.364 .975 9.00 187 4347 8.89 465 240 b) Machine at high speed (Test No. 10) 1620 1795 2820 1.220 .154 136C 540 47 CC 1050 3500 1450 "Be*} 1795 2820 1.320 .154 1360 500 4660 1050 3445 1250 1000 1800 2825 1.310 .1538 1364 333 4500 1045 3220 1350 In theoe tests the dynamic load wau 50?, greater than the static load at low speed and nearly 1607 of the static load at 'ni^h speec1. Thus the dynamic loads can no longer be neglected. The nuclcingham formula given rather close results at low speed, but gives a predicted value 250% largo.r than the measured load at high speed. i A further analysis of the Canadian Surnfier gears w i l l be made in the following pages. - 56 -TABLE VIII The Experimental Results of the • Canadian Sumner Gears. Machine at low speed (Test No. 9) 1 Meas. Meas.Max. Length of Torque at Equivalent max. Torque tooth load contact cycle nominal value tooth lcado lbs. in-lb lbs. in tooth space in-lb 2000 1000 1.08 2000 1000 1450 710 1.08 1500 735 1050 490 1.04 1000 470 560 240 1.04 500 214 High speed (Test No. 10) 1620 1450 1.10 2000 1790 1500 1250 1.10 1500 1250 1000 1350 1.07 1000 1350 — — — 500 — Machine at low speed ( Test No. 12) 2000 1020 1.10 2000 1020 1500 765 1.09 1500 765 900 450 1.08 1000 500 450 210 1.07 500 234 High speed (Test No. l l ) 2050 1200 1.07 2000 1170 1500 1300 1.07 1500 1300 1000 1500 1.07 1000 1500 —* 500 ~ Machine at low speed (Test No. 13) 1920 910 1.12 2000 950 1400 650 1.10 1500 700 1000 430 1.09 1000 430 450 230 1.08 500 255 High speed (Test No. 14) 2150 1300 1.12 2000 1200 1300 1200 1.12 1500 1380 1000 900 1.08 1000 900 — — — 500 — . 5.7 -Analysis of Load Cycle of Canadian Sumner Gears. For purposes of calculating the response of the system to a force Impulse function, consider an undamped linear single-degree of freedom system subjected to a load \jhich i s gradually applied and then maintained inde f i n i t e l y 9 , (Fig. 27b). y Fig. 21a F V Fe t ( Time.) Fi$. 21 b Fig. 21 c - 58 -If the applied force i s f ( t ) , the equation of motion becomes ,2 . ™ i p + K y = j (t) where ^ ( t ) = Fo t for 0<t<b b = 0 for t < 0 m* -yi where 2AJ"n = undamped angular frequency, K = spring constant m = mass of the system having a forced vibration. then d 2y ^ , .2 r, v —£. + y - j-(t) dt If at time t = 0, the i n i t i a l displacement f(0), and the i n i t i a l velocity J ( 0 ) are zero, then by method of Laplace Transformation Eq. becomes y<t> - Is- | w t . s i n ^ t ; -mb2»J^ :. ( n n ) . F o f t 1 1 y ( t> - Fo ff&Jt - sin&Tt) - f c ( t - b ) - sinW" n(t-b)] mb til 0 " I F mb Ifctfb + sinicT ( t - b ) - sin W t I n n n 59 [ s i n f e ^ C t - b ) - s i n ^ n t ] } when t ^  b F Q represents the static deflection of the system under the F action of a steady force F Letting _£ = Y . # Equations (2) and u j , S t * (3) can then be written to express the results in terms of a magnification or response factor y(t)/y st. — = £ " _L_ s l n W l t for 0 $ t $ b (4) -XSt) = l + _2_ CosWh<t-|) Sin ^ f o r t ? b ( 5 )  y s t ^ n b 2 2 The maximum value of the response ratio i s defined as the dynamic load factor. Then Dynamic Load factor = 1 + 2 S i n w n fc£b T ~ (6) Fig. 14b is the high speed tooth load pattern of the Canadian Sumner gears at a measured torque of 1620 in-lb ad - length of tooth load contact cycle = 1 > 1 3 " = tooth space. 1" be - length of one tooth carrying the load = 0.95 tooth space ab - the length for the load transferring to the tooth, or the time interval = 60 x . l l = .000275 sec. 1780 x 24 - 60 -cd - the length for the load drop or the time interval - 6 0 x - 0 7 " = 0.000175 sec. 1780 x 24 The calculated natural frequency of the vibration system formed by the gear and pinion and the mating teeth is about 3300 c/sec. The 13 formula derived by W.A. Tuplin is as follows: 7= J L 2-rr ( r l 2 + r 2 2 ) Natural frequency c/sec. r 2 2 I I M1 r2 M2 where i = k = Equivalent teeth stiffness ^ / i n . rl» r2 = Pitch radius in. 1^, 12 = Moment of inertia of rotating masses about their axes of rotation lb - in - sec Ml» ~ M a s s o f S e a r l b " sec 2/in. 2 In the formula a l l rotating masses are ignored except the gears themselves, and each gear is regarded as a ring with metal thickness under the teeth equal to the pitch of the teeth and radius of gyration as - 61 -equal to the pitch radius. The measured frequency of the tooth load pattern in Fig. 14b is 2860 cycles/sec, which agrees qualitatively with the calculated frequency. Therefore: WA = 2 IT/ = 2TT X 2860 rad/sec. b = 0.000275 sec (length ab in Fig. 14b). Wnb = 1/2 x 2 T x 2860 x 0.000275 - 2.47 rad. 2 Dynamic load factor = 1 + 2 x sin / tin b W„b \ 2 « 1.252 The steady force Fo from consideration of the torque i s ! , ^ 62ti6 - 560 lbs. 3 cos then F max. = Fo < 1 + 2 ^ sin Wnb - 560 x 1.252 - 700 lbs. Comparing the value of F m a x < found above with the measured maximum tooth load (1450 lbs), there is no satisfactory agreement. _ 62 _ Fig, 14b Tooth load pattern at a measured torque of 1620 in-lb at high speed (Test No. 10). Fig. 13b Tooth load pattern at a measured torque of 2000 in-lb low speed (Test No. 9). For analyzing the tooth load pattern of the Canadian Sumner Gears at low speed, a particular pattern at a measured torque of 2000 in-lb is shown in Fig. 13b. ad - length of tooth load contact cycle - i.07'71" ~ 1.07 tooth space, be - one tooth carrying the load «• 0.95 tooth space ab - length for the load transferring to the tooth, or the time interval - 6° x *? 7 - .00307 sec. 57 x 24 cd - the length for the load drop or the time interval » °° x 57 x 24 .0022 sec. - 63 -Taking the natural frequency as 2860 rad/sec then WV, = 2-n-f = 2ff x 2860 rad/sec b = .00307 sec. = 1/2 x 2 sr x 2860 x .00307 - 27.6. rad. 2 The steady force F Q « 2000 = 6 9 Q l b s < 3 cos 14%° The maximum force is therefore: . fe>_ b «= 865 lbs. Fm = F o | 1 * JL. * ^ W l l I <4,b 2 The value of the maximum force found above (865 lbs) agrees f a i well with the corresponding measured maximum tooth load (1000 lbs). - 64 -CHAPTER VI  CONCLUSIONS A. Accurately Cut Gears The test results indicate that in accurately cut gears the maximum load between teeth is not appreciably increased by dynamic effects. The load is shared by the teeth during the beginning and end of the contact cycle, and the period of load engagement of a given pair of teeth is quite close to the theoretical value. At high speeds, vibration is set up when the load i s transferred to a single tooth, the frequency of load vibration being that of the natural frequency of the tooth system. This vibration is the cause of the dynamic load effect. The test results show that the increase in tooth load due to this effect i s not very large. It appears that the Buckingham dynamic load equation greatly overestimates the additional load due to dynamic effects. In analyzing the dynamic tooth load, Buckingham assumed that a single pair of teeth are carrying the load at the c r i t i c a l phase of the load transfer. Hence, the Buckingham formula always gives larger values than actual measurements. In this connection D. W. Dudley commented: "Even though the Buckingham method may not always give an answer that w i l l agree with test data, i t is s t i l l the best method available at present". B. Form Cut Gears In form cut gears the tips and flanks of the teeth depart from the involute profile, and only about the middle third of the teeth is actually of involute profile. &s a result of this, the teeth do not share the load, - 65 -but only one p a i r of teeth carry the load at a time. There i s thus a sudden tr a n s f e r of load from one tooth p a i r to the other. At high speed t h i s sudden t r a n s f e r r e s u l t s i n a severe tooth load v i b r a t i o n . The added load due to dynamic e f f e c t s i s thus much larger than with accurately cut gears. Even though the dynamic load e f f e c t s i n these gears are larger than with accurately cut gears, the Buckingham dynamic load equation s t i l l overestimates the value of the' dynamic load. C. General Conclusions The tooth p r o f i l e shape and the accuracy in manufacture, i . e . the contact r a t i o , have profound influence i n the dynamic incremental forces. Buckingham*s equation always overestimates the value of the dynamic' tooth force e s p e c i a l l y i n the higher speed cases. The frequency of the dynamic load v a r i a t i o n i s close to the natural frequency of the v i b r a t i o n system formed by the gear and :the p i n i o n as masses and the mating teeth as a spring. The dynamic tooth load are l i n e a r l y proportion to the applied torque. Increasing the applied torque by 1007., the measured tooth loads also increase about 1007». " With a small applied torque, the length of the tooth contact cycle decreases s l i g h t l y , more v i b r a t i o n e f f e c t occurs and the maximum tooth load i s maintained f o r a longer period. In accurately cut gears, the speed does not e f f e c t the dynamic tooth -forces. The maximum tooth loads at high speed tests (approximately 2800 fpm p i t c h l i n e v e l o c i t y ) are very close to those at low speed te s t s (approximately 90 fpm p i t c h l i n e v e l o c i t y ) . The dynamic tooth - 66 -loads are very small and may be ignored. The question of dynamic loads on gear teeth has produced a large number of isolated observations and analyses. Much needs to be done in this f i e l d both analytically and experimentally, before a clear and exact picture as well as reliable predictions can be made of the complex dynamic load phenomena. - 67 -APPENDIX INSTRUMENTATION I. BAM-1 BRIDGE AMPLIFIER AND METER The BAM-1 Bridge Amplifier Meter is an instrument which can be ut i l i z e d for both, dynamic and static work with SR-4 strain gages, Statham pickups and similar resistance type transducers. The DC s t a b i l i t y of the meter is within 100 microvolts per hour. Frequency response is f l a t within 57o from DC up to 25,000 cps and slowly drops off, at higher frequencies. The DC transistor pre-amplifier has a gain of 150. For static measurements the meter on the BAM-1 reads tension to the right and compression to the l e f t of zero in the middle. The scales are arbitrary with a simple system of calibration to make them read directly in terms of units being measured. The bridge ci r c u i t is i n i t i a l l y balanced with the meter at zero. Thermoelectric effects are balanced out. For dynamic measurements the BAM-1 Bridge Amplifier meter was coupled to a DuMont 350 DC Cathode-ray Oscilloscope. II. DU MONT TYPE 350 CATHODE-RAY OSCILLOGRAPH The Du Mont Type 350 Cathode-ray Oscillograph is an instrument that can be used for many general applications. The high gain of this instrument assures that i t can be used directly with many types of transducers, while DC amplification provides that the lowest frequency portions of signals w i l l be faithfully reproduced. - 68 -BIBLIOGRAPHY 1. Buckingham, E., Analytical Mechanics of Gears« McGraw-Hill Book Company, Inc., New York, 1949. 2. Lasche, Oscar, Zeitchrift des Vereines, deutscher Ingenieure, 1899. 3. Flanders, Ralph E., Machinery, January, 1908 Issue. 4. Marx, Guido H., Strength of Gear Teeth, ASME Trans. Vol. 34, 1912, pp 1323, and MarK, G.. H. and Cutter, L. E., Strength of Gear Teeth. ASME Trans. Vol. 37, 1915. 5. Buckingham, E., Dynamic loads on Gear Teeth, ASME Research Publ. 1931. 6. Dudley, Darle W., Practical Gear Design, McGraw-Hill Book Company, Inc.^ New York. 7. Tuplin, W. A., Dynamic Loads on Gear Teeth, Machine Design, October, Vol. 25, 1953, pp 203. 8. Buckingham, E., Manual of Gear Design Section Two - Spur and Internal  Gears, Machinery, New York, 1935, pp 146-141. 9. Hartman, James B., Dynamics of Machinery, McGraw-Hill Book Company, Inc., New York, 1956, pp 226. 10. Reswick, J. B., Dynamic Loads on Spur and Helical Gear Teeth, ASME Trans., Vol. 77, 1955, pp 635. 11. Niemann, G., and Rettig, H., Dynamic Gear Tooth Forces (Dynamische Zahnkrafte Verein Deutscher Ingenieure, Dusseldorf, VDI-2 Bd, 99, Nr. 3, Seite 89, bis 128; Januar, 1957). 12. Utagawa, M., Measurements of Dynamic Loads on Gear Teeth. Proceedings of the 6th Japan National Congress for App. Mech., IV-14, 1956. 13. Tuplin, W. A., Gear-Tooth Stresses at High-Speed, Preceedings I, Mechanical Eng. Vol. 163, 1950, pp 162—175. - 69 -Other references not specifically referred to: Buckingham, E., Spur Gears, McGraw-Hill Book Company, 1928. Faires, V. M., Design of Machine Elements. Franklin, L. L., and Smith, C. H., Tests of Gear, ASME Trans., Vol. 46, 1924, pp 885. Lewis, Wilfred, Gear Testing Machine. ASME Trans. Vol. 36, 1914, pp 231. Merritt, H. E., Gears, London, S i r Isaac Pitman and Sons Ltd. Moore, M. B., Principles of Experimental Stress Analysis, Prentice-Hall Inc. New York, 1954. Nakada, T., and Utagawa, M., The Dynamic loads on Gear Caused by the  Varying E l a s t i c i t y of the Mating_Teeth, .Proceedings of the 6th Japan National Congress for App. Mech., IV-15, 1956. Perry, C. C , and Lissner, H. R. ,. The Strain Gage Primer, McGraw-Hill Book Company Inc., 1955. Schwamb, Merrill and James, Element of Mechanism. Semar, H. W. and McGinnis, R. E., Experimental Determination of  Gear Tooth Stresses in Large Marine Gears, ASME Trans. Vol. 80, 1958, pp. 195. Shigley, J . E., Machine Design, McGraw-Hill Book Company, 1956. Steeds, W.,. Involute Gears, Longmans, Green and Company. Timoshenko, S., Strength of Materials. Timoshenko, S., Vibration Problems in Engineering, D. Van Nostrand Company. Timoshenko, S. and Baud, R. V., Strength of Gear Teeth, Mechanical Eng. Vol. 48, November 1926, pp. 1105-1109. 

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