Page 472 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 13 Problems 459

13-42 Starting with the equations

7_^oo ^

and

Sxr(o)) = lim
^ T ^ o o 2777
= lim 7^—

T —00^ 77 /

*
show that
^FX ( ^ ) = ^i24>(oj)
^xf(^)
and
Sf((o) Sfx(^)
= H(ico)
^xf(^) ^f(^)
13-43 The differential equation for the longitudinal motion of a uniform slender rod is
;î2.
dt^ dx^
Show that for an arbitrary axial force at the end x = 0, with the other end x = I free,
the Laplace transform of the response is

u ( X ^___r (5/rX-v-/) I ^-(5/cX-v-/)]
sAE{\-e-^^^>'^Y J
13-44 If the force in Prob. 13-43 is harmonic and equal to F{t) = show that
co^[{iol/c){x/l - 1)]
w(x, t)
(oAE sin ((ol/c)
and
- sin [((ol/c)(x/I - 1)]
a{x,t) =
sin(o;//c) A
where cr is the stress.
13-45 With 5(co) as the spectral density of the excitation stress at jc = 0, show that the
mean square stress in Prob. 13-43 is
2 2t7 r* , \ • 2 ^
o- = nirj