The London, Edinburgh and Dublin philosophical magazine and journal of science . gard the displacements of the mass centre and elasticcentre as taking place along the same radius-vector fromthe axis of rotation, which likewise may be obtained fromthe general solution by putting b = 0. Let u be the ultimate amplitude of the vibration at theelastic centre, then maw2 . maco2 \/(c-mco2)2 + b2co2 ±{c-mw2)Put k — \J — = speed of free a) k, we have a u 1? 1 1 F CD2 F= : mco2(u — a) = 1 ma 1 CO2 and ¥ = ma>2(u — a)= -z —=mk2u. Plotting these results against the speed <w, we see that
The London, Edinburgh and Dublin philosophical magazine and journal of science . gard the displacements of the mass centre and elasticcentre as taking place along the same radius-vector fromthe axis of rotation, which likewise may be obtained fromthe general solution by putting b = 0. Let u be the ultimate amplitude of the vibration at theelastic centre, then maw2 . maco2 \/(c-mco2)2 + b2co2 ±{c-mw2)Put k — \J — = speed of free a) k, we have a u 1? 1 1 F CD2 F= : mco2(u — a) = 1 ma 1 CO2 and ¥ = ma>2(u — a)= -z —=mk2u. Plotting these results against the speed <w, we see that, 312 Prof. H. H. Jeffcott on Lateral Vibration of Loaded with a suitable alteration in scale, the same curves representboth u and F. The curves (fig. 4) have not been drawn close to theposition co — k, as in that region the damping prevents thevalues becoming indefinitely great, as would appear to bethe case if the formulae used for u and F were fully The curves exemplify clearly how that the critical speedis one that is particularly favourable to vibration, or one atwhich the shaft is very sensitive to lateral disturbing forces. If in the choice of the working speed the critical speed isBavoided by 10 per cent, on either side, i. e. if co does not liebetween 9k and llk, then the amplitude of vibration willnot be more than about five times the eccentricity of themass centre. If the shaft is in good balance this choice of workingspeed will therefore give good results and sufficiently steadyrunning. The better the balance the closer to the critical speed maythe shaft be worked. For while keeping the amplitude ofvibration u within a definite limit, we may approach k themore closely the smaller the value of a ; and further, thedamping will keep the amplitude rather lower than thatplotted. On the other hand, if the balance is bad the critical speedmust be avoided by a large margin, and it may even be Shafts in the Neighbourhood of a Whi
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Keywords: ., bookcentury1800, bookdecade1840, booksubjectscience, bookyear1840
