Page 365 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

352 Mode-Summation Procedures for Continuous Systems Chap. 11

Eq. (11.1-3), but by the equation
0 if j i
/ [m{x)<pj<p, + y(x)(A^.iA,] cbc ( 11.2-1)
M, if j = i
whieh ean be proved in the following manner.
For eonvenience, we rewrite Eqs. (9.6-5) and (9.6-6), ineluding a distributed
moment per unit length 2??(x, t):
A + ^ g ( ^ - < a ) = 0 (9.6-5)
dx
my A k A G \ ^ - ^ P{x,t) = 0 (9.6-6)
dx
For the foreed oseillation with exeitation p{x,t) and 3?î(x,/) per unit length of
beam, the deflection y{x,t) and the bending slope ijjix.t) can be expressed in
terms of the generalized coordinates:

'
> = E'?,(0<P;(-*')
(11.2-2)
•A = E'7/(0'A.,(-ï )

With these summations substituted into the two beam equations, we obtain
= E ^ , + kAG(<p'^ - ^2 -h 23î( jc, t)
(11.2-3)
m = 'LQi-jz{kAG{tp) - (Ay)) + p ( x , t )

However, normal mode vibrations are of the form
y = <p^{x)e“^'‘
(11.2-4)
Ip = 4>j(x)e'‘“i'
which, when substituted into the beam equations with zero excitation, lead to

^ (11-2-5)
- cojmipj = -j^[kAG{<p'j - lA)]
,
The right sides of this set of equations are the coefficients of the generalized
coordinates Qj in the forced vibration equations, so that we can write Eqs. (11.2-3)
as
= - Ei'yWyViAy ® i(x ,i)
( 11.2-6)
'H P j^Pj = - + p { x , t )