Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 640

Form the transformation matrix X [ x 1 x 2 x 3 ]. Use this matrix, along with M of Eq. (9) of

Example 10.1 and K of Eq. (11) of Chapter 5, Example 5.5, to compute M 1 , K 1 , C 1 , and F 1 ,
using Eqs. (10.133) through (10.136)

=

1 . 00000 0 0

M 1 = 0 1 . 00000 0 (5)

0 0 1 . 00000

10 4 10 4 10 4 3 . 42824 × − 3 . 99727 × 2 . 28076 ×

10 4 10 6 10 6 K 1 = − 3 . 99727 × 3 . 19139 × − 1 . 82093 × (6)

10 4 10 6 10 6 2 . 28075 × − 1 . 82093 × 2 . 71161 ×

0 . 000625 0 0

C 1 = 0 0 . 000625 0 (7)

0 0 0 . 000625

10 − 1 10 − 1 10 − 1 ]

F 1 = [0 . 77242 × − 0 . 90063 × 0 . 51388 × (8)

As expected, this condensed mass matrix M 1 , is diagonal. The condensed system equations
are given by Eq. (10.132), M 1 y + C 1 y ˙ + K 1 y = F 1 g ( t ). Integrate this relation and calculate

the displacements V using Eq. (10.131).
The Wilson method applied to the Ritz vector equations leads to the same results as

given in Example 10.11 for the Wilson

θ

θ method.

References

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