Page 425 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

412 Classical Methods Chap. 12

12-18 Using the Rayleigh-Ritz method, determine the first two natural frequeneies and
mode shapes for the longitudinal vibration of a uniform rod with a spring of stiffness
Â,) attaehed to the free end, as shown in Fig. PI2-18. Use the first two normal modes
:
of the fixed-free rod in longitudinal motion.
AE
-AA/U-|;
Figure P12-18.

12-19 Repeat Prob. 12-18, but this time, the spring is replaced by a mass as shown in
Fig. P12-19.

AE
= □ " ”0 Figure P12-19.

12-20 For the simply supported variable mass beam of Prob. 12-11, assume the deflection to
be made up of the first two modes of the uniform beam and solve for the two natural
frequencies and mode shapes by the Rayleigh-Ritz method.
12-21 A uniform rod hangs freely from a hinge at the top. Using the three modes 4)^ = a / / ,
(/>2 = sin (tta//), and (f)^ = x/I), determine the characteristic equation by
using the Rayleigh-Ritz method.
12-22 Write a computer program for your programmable calculator for the torsional system
given in Sec. 12.1. Fill in the actual algebraic operations performed in the program
steps.
12-23 Using Holzer’s method, determine the natural frequencies and mode shapes of the
torsional system of Fig. P12-23 when J = 1.0 kg • m^ and K = 0.20 X 10^' Nm/rad.

Figure P12-23.

12-24 Using Holzer’s method, determine the first two natural frequencies and mode shapes
of the torsional system shown in Fig. PI2-24 with the following values of J and K\
./j = J2 ^ 1.13 kg • m“

^4 = 2.26 kg • m'

K^= K, = 0.169 Nm/rad X 10'’
= 0.226 Nm/rad X 10'’