Advanced Methods of Structural Analysis

14.2 Free Vibrations of Systems with Finite Number Degrees of Freedom: Force Method 527

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a EI m 1 m 2 m 3

P 1 =1

P 2 =1

P 3 =1

M 1

M 2

M 3

3 l 16 2 l^16

l 16

l 4

l 8

l 16

2 l (^1616)
b
3 l
l 8
c
ml^3
w EI
1 = 4.9333
λ 1 = 31.5563
y 11 =1.0 y 21 =1.4142 y 31 =1.0
m 1 m 2 m 3
y 22 =0.0
y 12 =1.0
y 32 = –1.0
m 1 m (^2) m 3
ml^3
w EI
2 = 19.5959
λ 2 = 2.0
d
y 23 = –1.4142
y 13 =1.0 y 33 =1.0
m 1 m 2 m 3
ml^3
w EI
3 = 41.6064
l 3 = 0.44365
e
Fig. 14.10 (a) Design diagram of the beam; (b) Unit bending moment diagrams; (c) Mode shape
vibration which corresponds to fundamental (lowest) frequency; (d) Second mode of vibration;
(e) Third mode of vibration
Multiplication of corresponding bending moment diagrams leads to the following
results for unit displacements
ı 11 D
Z
M 1 M 1
EI
dxD
1
EI

1
2
l
4
3l
16
2
3
3l
16
C
1
2
3l
4
3l
16
2
3
3l
16

D
9
768
l^3
EI
;
ı 22 D
Z
M 2 M 2
EI
dxD
16
768
l^3
EI
;ı 33 D
Z
M 3 M 3
EI
dxD
9
768
l^3
EI
;