Page 348 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.8 Generalized Force Proportional to Displacement 335

From the second integral for Q,
1.200 - 0.100 -0.600 0.100
- 0.100 0.133 0 -0.0166
-0.600 0 0.600 - 0.100
0.100 -0.0166 - 0.100 0.0333
By adding the two matrices, the generalized force becomes
2.057 -0.150 -1.029 0.10643 V-
El i -0.150 0.2143 -0.01429-0.0262
I \ El -1.029 -0.01429 1.0286 -0.1643
0.1064-0.02612-0.1643 0.0571
El (
/3 ( El HV
It is now necessary to choose a numerical value for the rotation parameter
and combine the previous equation with the stiffness matrix. This was
done for rotation parameters 0, 1, 2, and 4 to obtain the computer results for the
eigenvalues and eigenvectors. Because the previous matrices fed into the computer
are those for the two-element beam with each element of length /, the eigenvalues
are those for a beam of length 21.
Examination of the eigenvalue expression indicates that for a beam of length
/ with each element of length //2, the length / must be replaced by 1/2 in the
TABLE 10.8-1 COMPUTER RESULTS FOR TWO-ELEMENT ROTATING BEAM OF LENGTH /

i A, for Beam of Length 21 Exact
El " ' / T mE
1 0.001841 3.51 3.515
0 2 0.07348 22.22 22.034
3 0.84056 75.15 61.697
4 7.08106 218.1 120.90
1 0.0035169 4.861
1 2 0.08445 23.82
3 0.86754 76.35
4 7.13759 219.0

1 0.0049532 5.77
2 2 0.095627 25.35
3 0.894749 77.54
4 7.19323 219.8
1 0.0103809 8.35
4 2 0.158008 32.58
3 1.04317 83.72
4 7.83817 229.5