Page 309 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

296 Vibration of Continuous Systems Chap. 9
9-14 Show that c = y¡G/p is the velocity of propagation of torsional strain along the rod.
What is the numerical value of c for steel?
9-15 Determine the expression for the natural frequencies of torsional oscillations of a
uniform rod of length / clamped at the middle and free at the two ends.
9-16 Determine the natural frequencies of a torsional system consisting of a uniform shaft
of mass moment of inertia with a disk of inertia / q attached to each end. Check the

fundamental frequency by reducing the uniform shaft to a torsional spring with end
masses.
9-17 A uniform bar has these specifications: length /, mass density per unit volume p, and
torsional stiffness IpG, where Ip is the polar moment of inertia of the cross section
and G the shear modulus. The end jc = 0 is fastened to a torsional spring of stiffness
K lb • in./rad, and the end / is fixed, as shown in Fig. P9-17. Determine the
transcendental equation from which natural frequencies can be established. Verify the
correctness of this equation by considering special cases for = 0 and K = oc.

Figure P9-17.

9-18 Name some of the factors not accounted for in the method presented in Sec. 9.4 for
the calculation of the natural frequencies of suspension bridges.
9-19 The new Tacoma Narrows Bridge, reopened on October 14, 1950, has the following
data:
/ = 2800 ft (between towers)
= 60 ft (width between cables)
k = 280 ft (maximum sag of cables)
w, = 8570 Ib/lineal ft
/() = ~P (rotational mass moment of inertia)

Calculate the new cable tension, the cable torsional stiffness, the new vertical and
torsional vibration frequencies, and compare with previous values. (Assume a reason­
able value for the radius of gyration p in determining the torsional mass moment of
inertia.)
9-20 The following data on the Golden Gate Bridge was obtained from reports provided by
the district engineer for the bridge:
/ = 4200 ft
h = 470 ft
6 = 90 ft
f
(O = 28,720 Ib/ft (total weight per lineal foot, including cables)