Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 590

28 . 82048 0 . 00000 0 . 00000 14 . 41024 0 . 00000 0 . 00000

0 . 00000 32 . 17792 − 15 . 70660 0 . 00000 11 . 05280 9 . 25193

0 . 00000 − 15 . 70660 9 . 96561 0 . 00000 − 9 . 25193 − 7 . 43177

m 1

= (4)

14 . 41024 0 . 00000 0 . 00000 28 . 82048 0 . 00000 0 . 00000

0 . 00000 11 . 05280 − 9 . 25193 0 . 00000 32 . 17792 15 . 70660

0 . 00000 9 . 25193 − 7 . 43177 0 . 00000 15 . 70660 9 . 96561

This mass matrix referred to the global coordinate system becomes

m 1 T 1 T m 1 T 1 =

31 . 33856 1 . 45381 − 13 . 60231 11 . 89216 − 1 . 45381 8 . 01241

1 . 45381 29 . 65984 − 7 . 85330 − 1 . 45381 13 . 57088 4 . 62597

− 13 . 60231 − 7 . 85330 9 . 96561 − 8 . 01241 − 4 . 62597 − 7 . 43177

= (5)

11 . 89216 − 1 . 45381 − 8 . 01241 31 . 33856 1 . 45381 13 . 60231

− 1 . 45381 13 . 57088 − 4 . 62597 1 . 45381 29 . 65984 7 . 85330

8 . 01241 4 . 62597 − 7 . 43177 13 . 60231 7 . 85330 9 . 96561

where T 1 is from Eq. (2) of Example 5.5.
For element 2: 10 − 2 0 ◦

= 3m, ρ = 24 . 96 kg/m, r y = 8 . 581 × m, α =

24 . 96000 0 . 00000 0 . 00000 12 . 48000 0 . 00000 0 . 00000

0 . 00000 27 . 88609 − 11 . 78524 0 . 00000 9 . 55391 6 . 93476

. − . . . − . − .

0 00000 11 78524 6 49180 0 00000 6 93476 4 83209

m 2

= (6)

. . . . . .

12 48000 0 00000 0 00000 24 96000 0 00000 0 00000

0 . 00000 9 . 55391 − 6 . 93476 0 . 00000 27 . 88609 11 . 78524

0 . 00000 6 . 93476 − 4 . 83209 0 . 00000 11 . 78524 6 . 49180

0 , m 2 m 2 Since α = = for element 2.

10 − 2 90 ◦ For element 3: = 3, ρ = 51 . 48 kg/m, r y = 8 . 915 × m, α =−

51 . 48000 0 . 00000 0 . 00000 25 . 74000 0 . 00000 0 . 00000

0 . 00000 57 . 52709 − 24 . 31006 0 . 00000 19 . 69291 14 . 29994

0 . 00000 − 24 . 31006 13 . 40137 0 . 00000 − 14 . 29994 − 9 . 96920

m 3

= (7)

25 . 74000 0 . 00000 0 . 00000 51 . 48000 0 . 00000 0 . 00000

0 . 00000 19 . 69291 − 14 . 29994 0 . 00000 57 . 52709 24 . 31006

0 . 00000 14 . 29994 − 9 . 96920 0 . 00000 24 . 31006 13 . 40137

This element mass matrix referred to the global coordinate system becomes

m 3 T 3 T m 3 T 3

=

57 . 52709 0 . 00000 24 . 31006 19 . 69291 0 . 00000 − 14 . 29994

0 . 00000 51 . 48000 0 . 00000 0 . 00000 25 . 74000 0 . 00000

24 . 31006 0 . 00000 13 . 40137 14 . 29994 0 . 00000 − 9 . 96920

= (8)

19 . 69291 0 . 00000 14 . 29994 57 . 52709 0 . 00000 − 24 . 31006

0 . 00000 25 . 74000 0 . 00000 0 . 00000 51 . 48000 0 . 00000

− 14 . 29994 0 . 00000 − 9 . 96920 − 24 . 31006 0 . 00000 13 . 40137

Follow the procedure outlined in Example 5.5, Eqs. (6) to (9), to assemble the global mass
matrix. The global displacement vector, for which the displacement boundary conditions