Page 381 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

368 Mode-Summation Procedures for Continuous Systems Chap. 11

and choose (p^ = siniTTx/l) and <^2 ^ l-O- Using Lagrange’s equation, show that
. . . 4 . . . 2 n

^22^2 “ ^
where
== it\ E I / M P )
= natural frequency of beam on rigid supports

0)22 —k / M
= natural frequency of rigid beam on springs
Solve these equations and show that
( / ? + l ) ± y ( / ? - l ) ^ + -^ R
^22 2
77^ — 8

Let yix, t) = [h + sm(7rx/l)]q and use Rayleigh’s method to obtain
= b = ( R - l ) T ^ / ( R - i r +
TT^

/? = M
I "2
A plot of the natural frequencies of the system is shown in Fig. Pll-12b.
11-13 A uniform beam, elamped at both ends, is excited by a concentrated force P^fit) at
midspan, as shown in Fig. PI 1-13. Determine the deflection under the load and the
resulting bending moment at the clamped ends.

Figure Pll-13.

11-14 If a uniformly distributed load of arbitrary time variation is applied to a uniform
cantilever beam, determine the participation factor for the first three modes.
11-15 A spring of stiffness k is attached to a uniform beam, as shown in Fig. PI 1-15. Show
that the one-mode approximation results in the frequeney equation
ME
tt^EI

Figure Pll-15.