Page 382 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 11 Problems 369

where
2 tt^EI
it)| = ---- ^
11-16 Write the equations for the two-mode approximation of Prob. 11-15.
11-17 Repeat Prob. 11-16 using the mode-acceleration method.
11-18 Show that for the problem of a spring attached to any point x = a of a beam, both
the constrained-mode and the mode-acceleration methods result in the same equa­
tion when only one mode is used, this equation being

(Í) = 1 + Mm

—
11-19 The beam shown in Fig. PI 1-19 has a spring of rotational stiffness K lb in./rad at the
left end. Using two modes in Eq. (11.3-8), determine the fundamental frequency of
the system as a function of K/Mw], where o), is the fundamental frequency of the
simply supported beam.

I, M
Figure PI 1-19.

11-20 If both ends of the beam of Fig. PI 1-19 are restrained by springs of stiffness K,
determine the fundamental frequency. As K approaches infinity, the result should
approach that of the clamped ended beam.
11-21 An airplane is idealized to a simplified model of a uniform beam of length / and mass
per unit length m with a lumped mass at its center, as shown in Fig. PI 1-21.
Using the translation of M,, as one of the generalized coordinates, write the
equations of motion and establish the natural frequency of the symmetric mode. Use
the first cantilever mode for the wing.

Figure Pll-21.

11-22 For the system of Prob. 11-21, determine the antisymmetric mode by using the
rotation of the fuselage as one of the generalized coordinates.
11-23 If wing tip tanks of mass M, arc added to the system of Prob. 11-21, determine the
new frequency.
11-24 Using the method of constrained modes, show that the effect of adding a mass
with moment of inertia /, to a point x, on the structure changes the first natural
frequency to, to

^ ^ M. ( - ^ i )

M,