Page 367 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

354 Mode-Summation Procedures for Continuous Systems Chap. 11

Figure 11.3-1.

loads is found from
3W = F{a,t) 8y{a,t) F M{a,t) 3y'{a,t)
= F{a,t)Z<P.(a) 8q, ^ M{a, t )Z¥. {a) (11.3-3)

Q, = = F{a,t)ip,{a) -h M{a,t)(p',{a)
Then, instead of Eq. (11.1-14), we obtain the equation

<7,(0 +OJjq¡{t) = ^ [F{a,t)ip^{a) + M{a, t)<p](a)] (11.3-4)

These equations form the starting point for the analysis of constrained structures,
provided the constraints are expressible as external loads on the structure.
As an example, let us consider attaching a linear and torsional spring to the
simply supported beam of Fig. 11.3-1. The linear spring exerts a force on the beam
equal to
F(a,t) = - k y ( a j ) = -k'£qj(t)<pj(a) (11.3-5)

whereas the torsional spring exerts a moment
M{a,t) = - K y \ a , t ) = -K'^qj{t)(p'j{a) (11.3-6)

Substituting these equations into Eq. (11.3-4), we obtain
1
Qi + «,9, - Jf -k(pXa)Y,qj<Pj{a) - Kip',{a)Y^qj(p'j{a) (11.3-7)
J j
The normal modes of the constrained modes are also harmonic and so we can
write
^7 -
The solution to the /th equation is then
1
^7 = -k(p,{a) Y.Qj<Pj(a) ~ Ftp',(a) J^qj(p'j(a) (11.3-8)
M,{coj -
If we use n modes, there will be n values of Qj and n equations such as the
preceding one. The determinant formed by the coefficients of the ^ will then lead