Page 302 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 9.7 System with Repeated Identical Sections 289

Because the system is linear, we can start with each boundary vector separately.
Starting with Cj, we obtain

yN
C/v =
V, N
Starting with we obtain

Vn
a£>/v = ^
y V . ,

These must now add to satisfy the actual boundary conditions at the terminal end,
which for a cantilever free end are
•A] Í
y _ y
> Cyy “1“ OlDj^
M 0
y ) N V 0 j
If the frequency chosen is correct, the previous boundary equations lead to
= 0
= 0
^NC
_ ^ N C _ _ _ > V
a = —
^ N D ^ND
which is satisfied by the determinant

NC
= 0
M ND V.
The iteration can be started with three different frequencies, which results in
three values of the determinant. A parabola is passed through these three points
and the zero of the curve is chosen for a new estimate of the frequency. When the
frequency is close to the correct value, the new estimate can be made by a straight
line between two values of the boundary determinant.

9.7 SYSTEM WITH REPEATED IDENTICAL SECTIONS

Repeated identical sections are often encountered in engineering structures. They
represent a lumped-mass approximation to the continous structure. For example,
the A-story high-rise building is often built with identical floors of mass m and