Page 310 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 9 Problems 297

Determine the cable stiffness Tb^ in torsion and compare with the old and new cable
stiffnesses of the Tacoma Narrows Bridge found in Prob. 9-19.
9-21 For a 1-DOF model of an airplane wing, assume an equation of the form

+ c6 kO = / , ( r , ^ ) +
where 6 is the angle of attack and r is the wind velocity. Discuss the possibility of
negative damping and the importance of the aerodynamic characteristics of the floor
and stiffness girders for suspension bridges.
9-22 Determine the expression for the natural frequencies of a free-free bar in lateral
vibration.
9-23 Determine the node position for the fundamental mode of the free-free beam by
Rayleigh’s method, assuming the curve to be y = sin(7rx//) - h. By equating the
momentum to zero, determine h. Substitute this value of h to find w,.
9-24 A concrete test beam 2 X 2 X 12 in., supported at two points 0.224/ from the ends,
was found to resonate at 1690 cps. If the density of concrete is 153 lb/ft \ determine
the modulus of elasticity, assuming the beam to be slender.
9-25 Determine the natural frequencies of a uniform beam of length / clamped at both
ends.
9-26 Determine the natural frequencies of a uniform beam of length /, clamped at one end
and pinned at the other end.
9-27 A uniform beam of length / and weight is clamped at one end and carries a
concentrated weight fF^ at the other end. State the boundary conditions and deter­
mine the frequency equation.
9-28 Solve Prob. 9-27 for the fundamental frequency by the method of equivalent mass
placed at the free end.
9-29 If a satellite boom of uniform weight IF/, is loaded with concentrated load IF, at

X = jc, and an end load IF,,, show that its fundamental frequency can be obtained from
the equation
3£/g
/'ÍF

where
y(xx)
W=W,, + 0.237 IF. + IF,
>’(/)
9-30 The pinned end of a pinned-free beam is given a harmonic motion of amplitude y,,
perpendicular to the beam. Show that the boundary conditions result in the equation
sinh ßl cos ßl - cosh ßl sin ßl
yi sinh ßl - sin ßl
which, for y,, 0, reduces to
tanh ßl = tan ßl