Page 361 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

348 Mode-Summation Procedures for Continuous Systems Chap. 11

In addition to T and U, we need the generalized force Q/, which is
determined from the work done by the applied force p{x,t)dx in the virtual
displacement 6q^.

SiV= dx

(11.1-9)

where the generalized force is

Qi = j'p{x,t)4>,{x) dx ( 11.1-10)

Substituting into Lagrange’s equation,

( 11.1-11)
dt [ dq^ ) ^q¡ dq^ ~
we find the differential equation for ^-(r) to be

Qi + ^ f p { x , t)d>^{x)dx ( 11.1-12)

It is convenient at this point to consider the case when the loading per unit
length p{x, t) is separable in the form

p{x,t) = - f p { x ) f { t ) (11.1-13)

Equation (11.1-12) then reduces to
Qi + <^hi = A ^ r , / ( i ) (11.1-14)

where

r, = J Jp(x)<A,(x) dx (11.1-15)

is defined as the mode participation factor for mode i. The solution of Eq. (11.1-14)
is then

Qi(t) = g,(0) COS o),i -f ¿ 9 ,(0 ) sin (o,t

+ I ^ U , / '/ ( 0 sin co,(t - i ) d i (11.1-16)

Because the iih mode statical deflection [with qft) = 0] expanded in terms of