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dynamic loads on the teeth of spur gears


DYNAMIC LOADS ON THE TEETH OF SPUR GEARS
By Stephen L. Harris, M.A., A.M.I.C.E.*
In a theoretical analysis of r single pair of high-class gean three i n t e d sources of vibration are considered: (1) Periodic variarions in the velocity ratio due either to running the gears above or below the designed load, or to small manufacnuing errors. (2) Period variation in the tooth stiffness. (3) Non-linearity in tooth stiffness. The amplitudes of vibrations caused by (1) depend on damping, while thoK caused by (2) or (3j wiii oniy occur if the damping is below a limit estimated at 0.07 of criiical damping. Damping currently reported for tests on steel gears is about 0.1 of critical, but might well be less. Dynamic tooth forces and amplitudes of vibrauon were found from photo-elastic stress panerns of model gears with both small and Large errors. The more accurate gears showed only vibration caused by (I), variation in the velocity ratio, because the damping was too great for the other modes. Information on damping is insuf5cient for satisfactory prediction of dynamic increments. If the damping in any g-ars fell to 0.07 of critical tht increment, instead of having a small value, might equal the applied load, irrespective of any errors, u l d this might prove disastrous.

INTRODUCTION

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A ~WMBER OF A ~ P T to measure dynamic loads have S been made since 1900. A large experimental programme was started in 1925 by a special research committee of the A.S.M.E., and their results were published in 1931. The loads on individual teeth in these experiments were deduced from consideration of the speeds at which the teeth first lost contact at given applied loads, but in the nature of the tests there could be sttle check on the accuracy of the deductions. Before this, however, Heymms and Kimball (1924)t had used photo-elastic models to find the dynamic stresses in a gear pinion. Considering the stage to which the technique was then developed it is less surprising that their results were inconclusive than that they obtained results at all. They gave exposures of up to 4 min with sparks in air as the light source, and studied speeds up to 1250 revlmin. More recently Niemann and Remg (1957) tested a number of steel gears, measuring the deflections of *e tip of o w tooth under both static and dynamic conditions. They give a considerable number of tooth loadings determined uperimentally with applied loads up to 3100 lb/in. face, both for
The M S . of this paper
u*u firsr reccivcd at the Imtitutiun a 251h March 1957, and in irs r m w d f m , as acaprui by the Cormdl for publication, on 26thjfune 1957. For a rrporr of the m r ~in , London, a t which this paper u w presented, see p. 112. Lecrurer in Engineering, Universip of Cambn.*r. An alphabetical lisr of refermcu L gima in A I ' :.

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gears with small errors, and for errors up to the very large value of 6.8 x 10-3 in. They also discuss the dependence of the static contact ratio on loading and on spacing erron. On the theoretical side there have been mmy papen discussing probable tooth loadings. The A.S.M.E. report (1931) contains a detailed analysis of the dynamic incranent under various conditions. Tuplin (1950) and Resaid 11954) have discussed the effect of various forms of assumed error. Strauch (1953) considered vibrations whch might b!dd up as a result of the continuous error baaran w o unconmed involute gears, but not in general terms. Zamn (1957) analysed the transient effects of four different f r s of om error, and also the steady dcct of ooe f o m of rcpatcd error. If the loads which might act ar r gear tooth were h o w n accurately, the next step in assusiq t)le suc?.gh of the tooth agamst athcr brei.: ..=a arcl.k 2 .c.aaledge . i of the mrrcspmding s u m dsnitb d bere the a d able upamad infonnttiao i mat a@ae Some of s the papers daling with this p b k m arc rcZhrrd t in o Appendix I. The mst noticeable gap in the a l evidence are that tbc only direct nvaurements d dynamic loads carried by individual t e c h are those just reported by Nicmann and Retrig, and there stan to be no measurements of the amplitudes of steady vibration of gears running u n d a

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