Page 341 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

328 Introduction to the Finite Element Method Chap. 10

i;/2-

@ Figure 10.7-1.
Solution: We use a single element (D - (D and first determine the inverse of the stiffness
matrix. Beeause r, = 0 , =0, the stiffness equation from Eq. (10.2-1) is
12 - 6/,
El
- 6/, 4/r

Its inverse, using the adjoint method, is
1 4/r 6/,
El 12/r
The equivalent hnite element forees, from Eq. (10.7-4), are

F^ = p4)y{x) dx = “ 31^^
' 1/2
-1 88 .
M2 = f ' - P<f>4iOI\d^ = - p l \ i ' { - f - + ^ ^ ) d ^ =
M/2 •'1/2 1536
J \ /I
d \ /I
Substituting these values into the inverted equation, we have
J_3
4/f - 32^/,
Ï
\2El 6/, 12 88 I
1536
' 52 528 ^ ( \
pit ^ 32 1536 5.125
\2EI 78 1056 48 E/ 7.000
32/, ^ 1536/,

These results agree with those ealeulated from the area-moment method.

10.8 GENERALIZED FORCE PROPORTIONAL
TO DISPLACEMENT

When the generalized force is proportional to the displacement, it can be trans­
ferred to the left side of the equation of motion to combine with the stiffness
matrix for the free vibration. Presented in this section are two cases: (1) for