Page 391 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

378 Classical Methods Chap. 12
iiV

Figure 12.1-7.

Solution: Referring to the table at the end of Chapter 2, we see that the deflection of the
beam at any point x (see Fig. 12.1-7) from the left end due to a single load IF at a
distance b from the right end is

The deflections at the loads can be obtained from the superposition of the deflections
due to each load acting separately.
Due to the 135-kg mass, we have

^ - ( 2 , , = - , u r | = 3.273 X f ».

>■3 - ’ll T I sei '^ *' - (3.0)“ - (1.5)“] - 2.889 x

Due to the 225-kg mass, the deflections at the corresponding points are
(9.81 X 225) X 2.5 x 3.0 10-'
6 X 5.5E/ [(5.5)^ - (3.0)^ - (2.5)^] = 7.524 El '

(9.81 X 225) X 2.5 x 1.5 10'
^'2 = "-------- 6X5.5EI----------■ (2-5)1 = 5-455 El
By adding y' and y'\ the deflections at 1 and 2 become

10' 10'
= 10.797 X - ^ m y^ = 8.344 X - ^ m
By substituting into Eq. (12.1-19), the first approximation to the fundamental fre­
quency is
9.81(225 X 10.797 + 135 x 8.344)E7
[225 X (10.797)^ + 135 x (8.344)1 lO"*

= 0.03129;/^ ra(j/s
If further accuracy is desired, a better approximation to the dynamic curve can
be made by using the dynamic loads rmo^y. Because the dynamic loads are propor­
tional to the deflection y, we can recalculate the deflection with the modified loads
gm^ and gm2iy2/y\)-