Page 237 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

224 Lagrange’s Equation Chap. 7

The velocity is
r{x) = x; 4>^{x)q,(t) (7.5-2)
/ = 1
and the kinetic energy becomes
N N
r = 2 X E <i,‘jjj4>,{x)(i)^{x) dm
i=\ /=!
N N
= ^ 2 ^ E E (7.5-3)

Thus, the generalized mass is

==/ 4>i{x)4>j{x) dm (7.5-4)
where the integration is carried out over the entire system. In case the system
consists of discrete masses, becomes

rri: E m^4>,{Xp)<i)j{x^) (7.5-5)
P= \
Generalized Stiffness (Axial Vibration)
We again represent the displacement of the rod in terms of the assumed modes
and the generalized coordinates:
n
0 = E <Pi{x)qi{t)
1
The potential energy of the rod under axial stress is found from Hooke’s law:
P _ P du
A ~ ^ dx
and the work done, which is.

dx (7.5-6)
Substituting for u{x,t) gives

U = J E AEip]<f>) dx
i 7

\ E Ekijq.dj (7.5-7)
/■ J