Page 344 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.8 Generalized Force Proportional to Displacement 331

generalized force associated with the load fi^xmdx, which is applicable to the first
element.
“0.4286 0.01429/ -0.4286 0.06429/ ■ fi'i'i
0.05714/2 -0.01429/ -0.009524/2 Ö,
{Ö,} = 0.4286 -0.06429/ <
0.02381/2
Example 10.8-3
Using one element, determine the equation of motion of the helicopter blade of
length / rotating at speed ii. Assume the blade to be rigidly fixed to the rotor shaft.
Solution: The mass and stiffness terms for the single element of length / are
-j (
156 22/ 54 -13/
ml 22/ 4/2 13/ -3/2 ml
420 < 420 MV
54 13/ 156 - 22/ ¿’2
-13/ -3 / - 22/ 4/2 t ^2 >
/
12 61 - 12 61
El 61 4/2 -61 2/2
- 12 -61 12 -61 Vi
61 2/2 -61 4/2 ö j
The term due to rotation is found from the generalized force Q given in Eq. (10.8-6).
For its evaluation, the integral involved is

'
'
til2/ ( X ( (pjipj dr ' dx = mEi^l f ^ [^(p](p':ld^ Id^
'
*() •'o •'() L*'o
where
1
<p\ = ( - 6 i + 6 ^ ^ )j
= 1 - 4^ + 3^2
'
<p, = (6^ - 6 f- ) \
^ ;= - 2 i + 3f2
Substituting these into the previous integral, we obtain the result
0.4286 0.01429/ 1 - 0.4286 0.6429/ iv^\
0.0142/ 0.05714/2 1 -0.01429/ -0.009524/2
mQ}l 1

- 0.4286 -0.01429/ 1 0.4286 -0.06429/ V2
- 0.06429/ -0.009524/2 1 -0.06429/ 0.02381/2 -