Advanced Methods of Structural Analysis

14.4 Free Vibrations of One-Span Beams with Uniformly Distributed Mass 539

Table 14.2

Comparison of the force and displacement methods for free vibration analysis

Force method (analysis in terms of displacements)

Displacement method (analysis in terms ofreactions)

Coupled differential

equations

1. Canonical form

ı^11

m

y 1

C 1

ı^12

m

y 2

C 2

:::

C

ı1n

m

n

yn

C

y^1

D

0

:::::::::::::ın1

m

y 1

C 1

ın2

m

2

y^2

C

:::

C

ınn

m

n

yn

C

yn

D

0

m

y 1

C 1

r^11

y^1

C

r^12

y^2

C

:::

C

r1n

yn

D

0

::::::::::::m

n

yn

C

rn1

y^1

C

rn2

y^2

C

:::

C

rnn

yn

D

0

2. Matrix form

FM

Y

C

Y

D

0

M

Y

C

SY

D

0

Type of coupling

Dynamical

Static

Matrices

Flexibility matrix

F

D

2 664

ı^11

ı^12

::: ı

1n

ı^21

ı^22

::: ı

2n

::: ::: ::: :::ın1

ın2

::: ı

nn

3 775

;

Y

D

2 66666

y^1 y^2 :::yn

3 77777

M

is diagonal mass matrix

Stiffness matrixS

D

2 664

r^11

r^12

::: r

1n

r^21

r^22

::: r

2n

::: ::: ::: :::rn1

rn2

::: r

nn

3 775

;

Y

D

2 66666

y^1 y^2 :::yn

3 77777

M

is diagonal mass matrix

Solution

Y

D

A

sin

.!t

C

'^0

/

Y

D

Asin

.!t

C

'^0

/

Equations for unknowns

amplitudes

A

i

1. Canonical form

m

ı 1

11



ı 1

!

 2

A

1

C

m

ı 2

12

A

C 2

:::

C

m

ın

1n

A

n

D

0

m

ı 1

21

A

C 1

m

ı 2

22



ı 1

!

 2

A

C 2

:::

C

m

ın

2n

A

n

D

0

:::::::::::::m

ı 1

n1

A

C 1

m

ı 2

n2

A

2

C

:::

C

m

ın

nn



ı 1

!

 2

A

n

D

0

r

11



m

! 1

 2

A

C 1

r^12

A

C 2

:::

C

r1n

A

n

D

0

r^21

A

1

C

r

22



m

! 2

 2

A

C 2

:::

C

r2n

A

n

D

0

::::::::::rn1

A

1

C

rn2

A

2

C

:::

C

r

nn



m

!n

 2

A

n

D

0

2. Matrix form

FM



(^12)!
(^) I
A
D
0
ŒS

!
2 M
A
D
0
Frequency equation1. Canonical form
2 664
m
ı 1
11

ı 1
!
2
m
ı 2
12
:::
m
ın
1n
m
ı 1
21
m
ı 2
22

ı 1
!
2
:::
m
ın
2n
:::
:::
:::
:::
m
ı 1
n1
m
ı 2
n2
::: m
ın
nn

ı 1
!
(^37752)
D
0
2 664
r^11

m
! 1
2
r^12
:::
r1n
r^21
r^22

m
! 2
2
:::
r2n
:::
:::
:::
:::
rn1
rn2
::: r
nn

m
!n
(^37752)
D
0

2. Matrix form

FM



(^12)!
ID
0
S

!
2 M
D
0