Page 378 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 11 Problems 365

Figure 11.5-2. First and second
First mode Second mode mode shapes.
The numerical values of the matrix [cl^j] and [b^^] from Eqs. (11.5-5), (11.5-6), and
(11.5-9) are
1.1774 2.6614
[« ,] = [C]’[m][C] =
2.6614 7.3206
7.200 10.800
[ M = [ C ]'[ ^ ] [ C ] =
10.800 19.200
Using these numerical results, we find the two natural frequencies of the system
from the characteristic equation of Eq. (11.5-12)

El
= 1.172

= 3.198
Ml^
Figure 11.5-2 shows the mode shapes corresponding to these frequencies.
Since Eq. (11.5-12) enables the solution of the eigenvectors only in terms of an
arbitrary reference, can be solved with = 1.0. The coordinates p are then
found from Eq. (11.5-9), and the mode shapes are obtained from Eqs. (11.5-1),
(11.5-3), and (11.5-4).

P R O B L E MS

11-1 Show that the dynamic load factor for a suddenly applied constant force reaches a
maximum value of 2.0.
11-2 If a suddenly applied constant force is applied to a system for which the damping
factor of the iih mode is ^ show that the dynamic load factor is given
approximately by the equation
Dj = \ cos (i)p
11-3 Determine the mode participation factor for a uniformly distributed force.
11-4 If a concentrated force acts at x = the loading per unit length corresponding to it
can be represented by a delta function / 5(x - a). Show that the mode-participation