Page 362 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 11.1 Mode-Summation Method 349

cf)^(x) is the quantity

A ( 0 = " ,/V (i) sin 0^,(1 - O (11.1-17)

can be called the dynamic load factor for the ith mode.
Example 11.1-1
A simply supported uniform beam of mass Mqis suddenly loaded by the force shown

in Fig. 11.1-2. Determine the equation of motion.
Solution: The normal modes of the beam are
(/>„(x) = V2 sm —^

co„ = {n^fy/EI/M,l^
and the generalized mass is

w • iniTX .
= -j- j 2sm^—j— dx = Mq
The generalized force is

j'^p{x,t)4>„{x) dx = g(t)J^^^^]/2 sin dx
0
n>ov/2 sm(mrx/l) x cos {mrx/l)
= 8i‘) rnr/l

-SKt) „„ cosnir

\/2/w'o

where g{t) is the time history of the load. The equation for is then

g{t)

1.0 —

0
(b)