Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 603

FIGURE 10.6
A uniform, simply supported beam.

Then the governing differential equation is

1 . 025 m u + 1 . 5 ku = 0 (9)

and, thus, the natural frequency is √ 1 . 4634 k / m with an error for the eigenvalue of

1 . 4634 − 1 . 46

= 0 . 23% (10)

1 . 46

EXAMPLE 10.5 Eigenvalues for a Beam Using a Consistent Mass Matrix Model
Use the displacement method to find the natural frequencies of the beam of Fig. 10.6. Model
the beam with two elements of equal length.
Begin by assembling the global stiffness and mass matrices. The element stiffness ma-
trix k i is given by Chapter 4, Eq. (4.12), while the consistent mass matrix m i is given by

Eq. (10.10a).

=

12 − 6 − 12 − 6

k aa k bb

k ab k bc

2 2

− 6 4 6 2

EI

k i

= = ( i = 1 , 2 ) (1)

3

− 12 6 12 6

k ba k bb k cb k cc

2 2

− 6 2 6 4

=

156 − 22 54 13

m aa m bb

m ab m bc

2 2

− 22 4 − 13 − 3

= ρ
420

m i

= (2)

54 − 13 156 22

m ba m bb m cb m cc

2 2

13 − 3 22 4