Page 379 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

366 Mode-Summation Procedures for Continuous Systems Chap. 11

factor then becomes = (/?,(«) and the deflection is expressible as
yU^t) E A ( 0

where cof = (pjYiEI/MP) and (/3,/) is the eigenvalue of the normal mode equation.
11-5 For a couple of moment Mq acting at x = a, show that the loading p{x) is the
limiting case of two delta functions shown in Fig. PI 1-5 as e 0. Show also that the
mode-participation factor for this case is
K; = l-
dx

\-d {x - a) ^ Six - a - e)
- X = 0 -------
i
C Figure PI 1-5.
11-6 A concentrated force Pq/U) is applied to the center of a simply supported uniform
beam, as shown in Fig. PI 1-6. Show that the deflection is given by
K,9,(x)
y(x,t) = - g ^ E
W ) "
2PoP s in ( ^ i) siniSTTy] siniSlTy]
El D M ------ X— r ^ D M + ■ - ■■
(37t ) ( 5 ^ r

J C " T 2L
Figure PI 1-6.

11-7 A couple of moment Mqis applied at the center of the beam of Prob. 11-6, as shown
in Fig. PI 1-7. Show that the deflection at any point is given by the equation
y(x,t) = - ^ E _ 3 A (0
(/3,/r
- sin sin (4i7y ) sin )
2Mo/^
El 3 -DiU) + -— 5 D4U) ^6(o +
( ztt) (47T) (67T)

Figure PI 1-7.

11-8 A simply supported uniform beam has suddenly applied to it the load distribution
shown in Fig. PI 1-8, where the time variation is a step function. Determine the