Page 423 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

410 Classical Methods Chap. 12

12-8 Repeat Prob. 12-7, but use the curve
y { x ) =yn,axx(' “ t )

12-9 A uniform cantilever beam of mass m per unit length has its free end pinned to two
springs of stiffness k and mass ruo each, as shown in Fig. P12-9. Using Rayleigh’s
method, find its natural frequency co^.

Figure P12-9.
12-10 A uniform beam of mass M and stiffness K = El/P, shown in Fig. P12-10, is
supported on equal springs with total vertical stiffness of k Ib/in. Using Rayleigh’s
method with the deflection = sinirrx/l) + b, show that the frequency equation
becomes

7^ k 4 ^T

M
7^ + — -y
2 77
By dco^/db = 0, show that the lowest frequency results when

, 77 / 1 KtT^^\ / 77 / 1 KtT^\
^ ~ ~ J \ 2 ^ ^ i T j - V 2k

k/2> Sk/Z
Figure P12-10.

12-11 Assuming a static deflection curve
y ( x ) = y „ 0 < JC<

determine the lowest natural frequency of a simply supported beam of constant El
and a mass distribution of
t(x) = m „ j(l - j )

by the Rayleigh method.