Page 578 - Aircraft Stuctures for Engineering Student

13.3 Structural vibration 559

In this problem the mass possesses an inertia about its own centre of gravity (its
radius of gyration is not zero) which means that in addition to translational displace-
ments it will experience rotation. The equations of motion are therefore
rnijSll + rnr’ijb,, + u = o (9
mij521+ rnr2e;SZ2 + e = o (ii)

where u is the vertical displacement of the mass at any instant of time and e is the rota-
tion of the mass from its stationary position. Although the beam supports just one
mass it is subjected to two moment systems; M1 at any section z due to the weight
of the mass and a constant moment M2 caused by the inertia couple of the mass as
it rotates. Thus

Hence

from which

413 21 312
511 =E’ 522 =E’ 512 = 521 =-
2EI
Each mode will oscillate with simple harmonic motion so that

v = vo sin(wt + E): e = eo sin(wt + E)
and
e=-&
2
Substituting in Eqs (i) and (ii) gives

(1 - u2rnG)u - u-mr 2 -e 312 = o
7
2EI (4

(vii)