. The strength of materials; a text-book for engineers and architects. f the angular velocity is w . •. Centrifugal force = F = —^ 9Ai . F^ ? T? 48EI8^^^^=48EI •?^- p 48EI8 Wco^, , ,/48EI W(o2^ Wco^e l^ 9 y 9 W 0)2 e 48 E I .8 = 9 l^ 9 4:SElg -w^WP ^^^ From this equation it is clear that 8 will become indefinitelygreat if 48 E I (/ - w^ W Z^ = 0 ^.6. Z^^LX (2) This value of w gives the critical speed. For mild steel E = 30 x 10^ lbs. per sq. in. and g =32*2 X 12 ins. per sec. per sec.; if therefore W is in lbs. andI and I in inch units 48 X 30 X 106 X 32-2 x 12 .1 WP = 746,000. / radians
. The strength of materials; a text-book for engineers and architects. f the angular velocity is w . •. Centrifugal force = F = —^ 9Ai . F^ ? T? 48EI8^^^^=48EI •?^- p 48EI8 Wco^, , ,/48EI W(o2^ Wco^e l^ 9 y 9 W 0)2 e 48 E I .8 = 9 l^ 9 4:SElg -w^WP ^^^ From this equation it is clear that 8 will become indefinitelygreat if 48 E I (/ - w^ W Z^ = 0 ^.6. Z^^LX (2) This value of w gives the critical speed. For mild steel E = 30 x 10^ lbs. per sq. in. and g =32*2 X 12 ins. per sec. per sec.; if therefore W is in lbs. andI and I in inch units 48 X 30 X 106 X 32-2 x 12 .1 WP = 746,000. / radians per sec (3) 74,600 X 60 /nr 2 TT MWP = •711 X W^L revolutions per minute (4) For a round shaft of diameter d inches we have I ird^ 64 •158 X 10^ d-^ n = j-^=^^— 564 THE STRENGTH OF :VIATERIALS Working from the transverse vibration we have, as on p. 337, ^ = 2 77 / Weight ^ g X force to cause unit displacement 48 E I . •. Force = t = 2- V48EI^ for unit deflection Frequency 1 1 t = o-a/—^-^ per second 2-V WZ3 ^ -^n J^^^V^T X 60 \ WZ^. Fig. 263. This,, for the given value of E, is exactly the same result asis obtained in equation (4) above. If the critical speed is exceeded either by providing guideswhich prevent the excessive deflection or by speedmg up soquickly that the inertia of the shaft prevents the dangerousdeflections from developing, the shaft will settle down and nui smoothly in a deflected form (Fig. 263), the weightrotating about an axis which gradually approaches its centreof gravity as the speed increases. This fact is made use of inthe flexible shaft of the De Laval turbine. If t-j is the critical velocity we may put in equation (1) ROTATING DRUMS, DISKS, AND SHAFTS 565 i. e. 8 = WP -WP o.;2 (u2 — O) 2 •e (5) This gets numerically less as o) increases, so that as the speed increases more and more the shaft tends to straighten out. If the shaft is horizontal and the weight is perfectly balanced, W Pand there is an initial
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