Page 421 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

408 Classical Methods Chap. 12

equation of the form
-v\
M
e ( 12.12-6)
M 2 M 4 y
where matrix [u] is the product of all the transfer matrices of the structure.
The advantage of the transfer matrix lies in the fact that the unknown
quantity at 1, i.e., 6^, for the cantilever beam, need not be carried through each
station as in the algebraic set of equations. The multiplication of the 4 x 4
matrices by the digital computer is a routine problem. Also, the boundary equa­
tions are clearly evident in the matrix equation. For example, the assembled
equation for the cantilever beam is
Í v \ -— — — —
-

-

M 1 — — — — 0 (12.12-7)
0 — — W 3 3 W 3 4 < 0
1 0 j„ W ^44 iij

43
and the natural frequencies must satisfy the equations
0 = 1/33^ + W34
0 = 1/43^ + W44
or
M 4
; + W44 == 0 ( 12.12-8)
^33
In a plot of vs. co, the natural frequencies correspond to the zeros of the curve.
P R O B L E MS

12-1 Write the kinetic and potential energy expressions for the system of Fig. P12-1 and
determine the equation for by equating the two energies. Letting X2/x^ = n, plot
0)^ versus n. Pick the maximum and minimum values of co^ and the corresponding
values of n, and show that they represent the two natural modes of the system.

3/f 2k
— V\AA/—I iTi I—W\A/— 2 —W\A/—
Figure P12-1.
12-2 Using Rayleigh’s method, estimate the fundamental frequency of the lumped-
mass system shown in Fig. P12-2.