040 Dr. C. Chree on the Whirling and 
§ 43. When there is no joad, and an algebraic type of 
vibration is assumed, the application of the Lagrangian 
equations is even simpler. Taking, for example, the case of 
a shaft supported at both ends, we have for the displacement, 
(cf. (6) case 2) 
ysnn(l — Qe? 4.0) 4 | naa 
whence 
ie °) 
T=tlop | (P+ ’y)\de=top(7? + o'n’)| a? (BP —2la? + a? Pda 
0 0 
ry, 
= F(" + 077”) (31/630)opP,. . (18) 
l bath 
V=1EI) (d2y/de?)P?de= LBL | 144.0°(«—iPde 
0 0 
= 30°( 24/5) HP, ho 4h 
Lagrange’s equation 
dat al. dV 
didn © dy dy 
gives 
(7 — wn) (31/630)ap 1° + (24/5) nEI? =0. 
Assuming 9 « cos kt, and so _7/n=—k*, we have (ef. (6), § 12) 
gt yl DA DJG TE iy coe ess Be, d 
§ 44. Under certain circumstances an equation of type (1), 
§ 1, may be shown to be true for vibration frequencies. The 
ordinary differential equation for africtionless simple harmonic 
motion is | 
Mad 2a/dt2-2= Nir ==.0). 5 oles eee (21) 
where M isa quantity of the nature of a mass, and F a force 
of restitution, such as is exerted by a spring. The frequency 
k/27 of the corresponding vibration is given by 
i = OUT ict 0 ene 
Suppose, now, that the force of restitution remains the 
same whether we apply one or a series of loads, M,, M,, &e. 
When the loads are put on one at a time, the corresponding 
frequency equations are 
k= G/F; Leo? = Mey e atin (23) 
when put on all together we have for the frequency equation 
1/k?=(M,+M,+... )/F 
= 1/ky? +1/k7 +. ce . * . ° . (24) 
