Transverse Vibrations of Rotating Shafts. d31 
When the pulley is close to a support it exerts—e/7. the 
analogous case in § 17—an exceedingly small effect on the 
critical velocity in the practical case of shaft and pulley 
combined. Except when the pulley is close to a support, we 
see in Table V1. a close agreement between even the simplest 
formula (19) and (18). The more complicated formula (23) 
agrees pretty closely with (18) even for the values 1/16 tad 
15/16 of c/a; for the other values the two would be in 
practical agreement. 
§ 32. Table VIL. compares the critical number of revolutions 
actually observed by Dunkerley for the case of equal spans, 
with those which he calculated from (2) and (18), and the 
corresponding results from (27), whick altogether neglects I’. 
TABLE VII. 

C/@ «..--- 9 tio. 1 eee 2: Precis 15/16. 
| 
| 
Bias | | 
| 
| 
| 

oN Ge On a ee ae 2) 
fe | Bee | 







| Observed | ‘4430 4524 | 3930 | 3213 | ot 4 | 2600 | 3846 [odes 4402 | 4220 
) Dunkerley | | | | 
ifrom (2)&(18)| 4440 | 4411 | 3925 3286 3334 | 267] | 3657 | tod 441] | 4362 

et (27)..., 4438 | 4381 | 3967 3516 | 3604 fee 3916 #498 [3881 
| 
| 
As theory led us to expect, the ee of (27) with 
observation is not so good for the heavier pulley II. as for 
the lighter. 

Case 6 (Dunkerley’s Cases VI. and XIV.). 
§ 33. Shaft fixed in direction at both ends. 
UML LL 
ULL, A Cc 3 WZ 
(a) Unloaded shaft : Euler-Bernoulli method (Dunkerley, 
l. c. p. 298). 
Measuring « from an end A, and denoting AB by J, we 
have 
y=a4 (cosh wr — cos ww)(sinh wl— sin pl) 
— (sinh wx—sin wx)(cosh wl—cospl)}, . . (1) 
where « is a constant, and the value of w is given by 
cosh wl coswl—1=0. . . . . . (2) 
20 2 
